Physics 160 Lecture Notes

Professor: Mikhail LukinNotes typeset by Emma Rosenfeld and Mihir Bhaskar

February 24, 2021

Contents

1 Introduction 21.1 Types of quantum technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Quantum foundations 42.1 States, Measurements and Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Evolution of quantum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Quantum bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Quantum operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 Generalized evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.9 Examples - quantum channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.9.1 Depolarization channel of the qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9.2 Dephasing channel of the qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.10 Master equation for density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.10.1 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.10.2 Density operator evolution for Markovian environments . . . . . . . . . . . . . . . . . . . 152.10.3 Example: spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.11 Generalized measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.11.1 Example: POVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.12 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.12.1 Example: Bell states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.12.2 Example: multiple qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.13 Properties of Bell states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.14 Schmidt Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.15 Unitary evolution of entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.16 Information content of Bell States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.16.1 Einstein’s Principle of Locality & Hidden Variable Theory . . . . . . . . . . . . . . . . . . 242.16.2 Bell inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.16.3 Violation of Bell’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.16.4 Loopholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.16.5 Applications of Bell’s inequalities to quantum information processing . . . . . . . . . . . . 26

2.17 Applications of entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.18 Mixed state entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.19 Multipartite entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.19.1 GHZ states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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2.19.2 W states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.19.3 Cluster states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.19.4 Remarks about multipartitle entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . 322.19.5 Tensor network states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Quantum algorithms 333.1 Simple quantum algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Quantum parallelism example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.2 Deutsch’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.3 Deutsch-Jozsa algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.4 Bertstein-Vazirani problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.5 Summary: simple quantum algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Simon’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Quantum search algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Quantum Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.1 Discrete Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.2 Quantum Fourier Transform definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Quantum phase estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.6 Order finding and factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.7 Implementing quantum algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.7.1 Universality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.7.2 Discrete set of universal operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.7.3 Other approaches to quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Implementation of quantum computers 544.1 Neutral atoms and ions: background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Trapped Ion Quantum Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Approach to two-qubit operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Neutral atom quantum computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Superconducting quantum computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Quantum error correction 675.0.1 Classical error correction: review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1 QEC key idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.1.1 Bit flip error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.1.2 Phase errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.1.3 Shor’s 9 qubit code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.1.4 Implementing QEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.1.5 Example: FTQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6 Quantum complexity theory 71

1 Introduction

There is vast experimental evidence that quantum mechanics is complete and correct. Quantum theory is now thebasis for many ubiquitous technologies, such transistors and lasers. We also understand why many objects aroundus don’t behave quantum mechanically: measurement collapses quantum states into one observable outcome oranother (for example, dead or alive in the famous case of Schrodinge’s cat). In fact, the measurement doesn’thave to be active. In practice, physical systems aren’t isolated from their environment, and the interaction withthe environment has the effect of making many measurements on the systems, such that the dynamics convergeaccording to classical laws of physics.

At the same time, information is closely connected to physics - to acquire information one needs a physicaldevice, to store information one needs a physical system, and to process it one executes physical operations onbits. Is there a limit to information and information processing? A bit of information in the classical world iseither 0 or 1. As Moore’s law continues, a transistor will become the size of one atom. What does that mean

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for the classical bit? Can one encode a bit of quantum information in a quantum system, such as an atom,nucleus, or photon? Can we use quantum mechanical evolution to perform computation? In this case, what isthe speed and performance? Is there a fundamental limit to Moore’s law? Is this a new opportunity? Theseare the questions we seek to answer in the field of quantum information processing. It will rely on two centralconcepts:

• Superposition: the idea that a particle can be in multiple states at the same time. The canonical exampleis Schrodinger’s cat, in which a cat is in a superposition of dead and alive.

• Entanglement : the idea that objects in superposition can be linked together and contain mutual informa-tion, even if they are physically separated from each other. Einstein argued in the famous EPR (Einstein-Podolsky-Rosen) paradox1 that quantum mechanics is not complete, because affecting one particle at oneside would affect the other, even if they are separated by an arbitrary distance.

1.1 Types of quantum technologies

Motivated by the questions defined above, physicists have identified an opportunity to use quantum physics toprocess and store information in three specific areas.

1. Quantum Metrology. Using the concepts of superposition and entanglement, one can make measurementsthat’s much more precise than any classical device. However, these superpositions are generally fragile.

2. Quantum Communication. Quantum communication uses the concepts of superposition and entanglementto transmit information in a secure way. If a third party, an adversary, tries to measure the superposition,they leave a trace because the measurement affects the state. One can therefore detect eavesdroppers,enabling fundamentally secure communication channels.

3. Quantum Computing. Suppose one can prepare a register in a superposition of input states, and utilizethem in a machine which can take the superposition as input, and then quantum mechanically performquantum logic (for example, additions, subtractions, etc.). These different inputs interfere with each otherthroughout the algorithm. This provides the possibility of doing something which is called ‘quantum par-allelism’, which uses interference to perform computations. These devices can potentially have much morecomputational capabilities than classical computers. They are also useful for understanding and simulatingthe behavior of quantum systems. For example, if one has a many body system from many interactingsubsystems and lets it evolve, the resulting state and dynamics would be very challenging for classical com-puters to simulate. This is the basis for quantum complexity, and an opportunity of quantum simulation,which is a subset of quantum computing.

However, the field faces a number of serious challenges:

• Nobody knows how to build truly large scale quantum machines. The largest are only 10’s of qubits,and how to bridge this gap is not clear. The main problem is that the qubits are coupling to envi-ronment. Theoretically, there are some concepts like quantum error correction and fault tolerance.We know theoretically we should be able to build these machines, but the necessary hardware is wellbeyond current imaginable experimental capability.

• It isn’t yet clear what quantum computers will be useful for. This is an algorithm challenge. It ispossible to build useful algorithms which utilize a quantum advantage?

Nevertheless, now is a special time. In several labs, quantum machines of increasing complexity are beingbuilt. For example, 50 or so qubits can maintain superposition and are programmable, similar to the timesof 1940’s and 1950’s for classical computers. The physics of maintaining entangled states and utilizingthem is also interesting in general, because it poses an exciting opportunity for science and technology.This course will aim to cover a foundation of quantum information science, and bring the student close tothe forefront of the field. There will be three components:

1See Einstein, Podolsky, Rosen, Physical Review 47, 777 (1935) for the original paper, as well as the “resolution” to this paradox:Bell, Physics Physique Fizika, 1, 3 (1964).

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(a) Quantum foundations, focusing on physics of open quantum systems; understanding how they evolve;how the dynamics are controlled; understanding entanglement.

(b) Building and using quantum machines, focusing on quantum computers; basics of quantum algorithmsand how to implement them; discussion of how the most advanced quantum machines are built.

(c) Connections to practical quantum information systems; students will simulate quantum systems, tofurther understand why it’s hard to do so on classical systems; explore near term quantum computingmachines, which are now available as a service; employment of a combination of classical numericsand web-based quantum computer access.

Preskill’s lecture notes will form the basis of the course, as a high-level undergraduate or introductory levelgraduate class. A little bit of programming experience will be helpful.

2 Quantum foundations

2.1 States, Measurements and Observables

How do we describe states, measurements and evolution? Here we review some basics of atomic physics andquantum mechanics.

References: Preskill’s notes, Nielsen & Chaung, McMahon

2.1.1 States

States are a vector in an n-dimensional space, called a Hilbert space H . We will use Dirac notation for a statewith a ket by |ψ〉 in a given basis, specified by n complex amplitudes, a1, a2, ..., an:

|ψ〉 =

n∑k=1

ak |k〉 (1)

Where |k〉 = (0, 0, 0, ..., 1, 0)T has a 1 at the kth component, and zero elsewhere.The bra is given by:

〈ψ| =n∑k=1

a∗k 〈k| (2)

The inner product 〈φ|ψ〉 is a complex number given by:

〈φ|ψ〉 =

n∑k=1

c∗kak (3)

With the important properties:|〈φ|ψ〉 |≤ 1; and normalization: 〈ψ|ψ〉 = 1.

2.1.2 Operations

Operations take states of physical systems and convert them to other states. The operator A is defined as:

A |ψ〉 = |φ〉 (4)

Such that the following matrix A:A = |φ〉 〈ψ| (5)

is a nxn-dimensional matrix if |φ〉 and |ψ〉 are in H .

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2.1.3 Observables

An observable is a property of the physical system such that it at least in principle can be measured. It isrepresented by a Hermitian matrix A, which means that it is equal to its complex transpose.

For all Hermitian operators, there is a spectral decomposition - one can find a basis in the Hilbert space wherethis matrix will be diagonal:

A =∑n

anPn (6)

Where an are eigenvalues and Pn = |n〉 〈n| are a complete set of Hermitian operators called projectors such that:

PnPm = δn,mPn (7)

The set of projects are a complete set such that they span the entire Hilbert space:∑n

Pn = I. (8)

2.1.4 Measurements

The outcome of measurements of the observable A for a system in an arbitrary state |ψ〉 will be one of theeigenvalues an with probability pn = 〈ψ| Pn |ψ〉 and state Pn |ψ〉 /

√pn. The outcome of the measurement

is generally probabilistic.

Remarks (the following is true for any closed quantum system):

1. Repeated measurements will yield the same results as the first measurement, hence ‘projection’.

2. One can measure the expectation value, or average value of the observer by repeated measurements afterre-preparing the superposition and measuring many times:

〈A〉 = 〈ψ| A |ψ〉 =∑n

anpn (9)

3. If you are given one copy of the unknown state ψ, you will not be able to determine the state with a singlemeasurement. A single measurement on the state does not reveal complete information about it. This isthe basis of quantum cryptography.

2.2 Evolution of quantum states

Dynamics are given by the Schrodinger equation:

ihd

dt|ψ〉 = H |ψ〉 (10)

The solution can be obtained by integrating with respect to time (assuming the Hamiltonian is time independent):

|ψ(t)〉 = U(t) |ψ(0)〉 , (11)

where:U(t) = e−iHt/h (12)

The operator U(t) is unitary since it is easy to see that U(t)−1 = U(t)†.

Remarks:

1. To clarify what we mean by an exponential of a matrix, we make explicit the following definition of operatorfunctions

f(A) =∑n

cnAn (13)

For an operator A.

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Figure 1: A qubit representation of a Bloch vector on the Bloch sphere. Picture from Wikipedia.

2. The unitary evolution preserves the norm of the state:

〈ψ(0)|ψ(0)〉 = 〈ψ(t)|ψ(t)〉 = 1 (14)

3. Unitary evolution is time reversible. By sending t→ −t, the state |ψ(t)〉 returns back to its original state.

4. Unitary evolution is both linear and deterministic. This is to be contrasted with measurement, which isfundamentally probabilistic.

2.3 Quantum bits

A qubit is composed of two state systems in a Hilbert space H ≡ |0〉 , |1〉. Free particles and harmonic oscil-lators cannot be approximated as quantum bits.

Let’s discuss the example of a spin 1/2 system as a qubit. The most general state can be written:

|ψ〉 = c0 |0〉+ c1 |1〉 (15)

Where |c0|2+|c1|2= 1 for normalization. One can parameterize the qubit using two angles θ and φ, rewriting thestate as:

|ψ(θ, φ)〉 = cos θ/2 |0〉+ eiφ sin θ/2 |1〉 (16)

The phase φ determines the phase between the two components, and the angle θ determines the probabilities offinding the states in |0, 1〉. This representation is convenient because one can represent it on the Bloch sphere.The Bloch vector is a unit vector in 3D, defined such that:

|0〉 ≡ |↑〉 , |1〉 ≡ |↓〉 . (17)

Measurement along the z-basis corresponds to projection along the up or down directions. See figure 1 for avisualization.

The spin matrices Sx,y,z are observables and are equal to the Pauli matrices (up to a factor of 1/2):

σ1,2,3 = σx,y,z = X, Y , Z (18)

Remarks:

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1. Properties of Pauli matrices:σ2α = 1 (19)

So the eigenvalues are ±1.

2. The Pauli matrices are traceless:Trσα = 0 (20)

3. One can write the Pauli matrices as:σz = |0〉 〈0| − |1〉 〈1| (21)

σx = |0〉 〈1|+ |1〉 〈0| (22)

σy = i |1〉 〈0| − i |0〉 〈1| (23)

4. The eigenstates are: Z : |0〉 , |1〉, X : (|0〉 + |1〉)/√

2, (|0〉 − |1〉)/√

2 ≡ |+x〉 , |−x〉, and Y : (|0〉 +i |1〉)/

√2, (|0〉 − i |1〉)/

√2 ≡ |+y〉 , |−y〉.

On the Bloch sphere, the states |±x〉 correspond to vectors pointing along the ±x direction, and the samefor y.

5. The commutation relation holds:[σα, σβ ] = 2iεαβγσγ (24)

such that the product of two Pauli matrices is another Pauli matrix.

6. The set σx, σy, σz,1 forms a complete basis for the 2x2 matrices: any 2x2 operator can be expressed asa linear combination of these operators.

2.4 Quantum dynamics

Considering a general form of a Hamiltonian:

H =h

2

3∑i=1

σi ≡hω

2σ~n (25)

Where ω =√

(∑h2i ), ni = hi

ω such that ~n is a unit vector. The vector ~n can be associated with the direction ofa magnetic field, where the Bloch vector is the spin that precesses around the field.

Quantum mechanically, we can describe this precession through the unitary evolution:

U = e−iHt/h = 1 cosωt

2− i sin

ωt

2~n · ~σ (26)

For example, let ~n = z. Then U = 1 cos ωt2 − i sin ωt2 σz. Multiplying an original state |ψ(θ, φ)〉 by this unitary,

we can solve for its time evolution:

|ψ(t)〉 = e−iωt/2(cos θ/2 |0〉+ eiφ+iωt sin θ/2 |1〉) (27)

These dynamics correspond to the Bloch vector rotating about the z axis on the Bloch sphere, at frequency ω.More generally, the vector ~n is the direction of the magnetic field that the Bloch vector rotates around on theBloch sphere.

Remarks

1. Probability amplitude method. Define the ansatz:

|ψ(t)〉 ≡ a0(t) |0〉+ a1(t) |1〉 (28)

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The time dependence is fully encoded in the coefficients. By plugging into the Schrodinger equation, werecover a system of two linear differential equations for a0,1:

ida0

dt=h3

2a0 +

(h1 − ih2)

2a1,

ida1

dt=−h3

2a1 +

(h1 + ih2)

2a0

(29)

For example, for h1 = h2 = 0, there is free precession about the z axis. For h3 = 0, we get two equations:

d2a0

dt2= −iΩda1

dtd2a1

dt2= iΩ∗

da0

dt

(30)

Where Ω ≡ (h1−ih2)2 . The solutions are sines and cosines, which is what we call Rabi oscillation about the

axis given by ~n.

2. Time dependent fields. Reference: AMO II, physics 285b lecture notes

For a time dependent Rabi frequency, we can incorporate the time dependence as a retarded time using achange of variables:

dτ ≡ dtΩ (31)

Then the solution is sines and cosines of the integral∫

Ωdτ . To flip the state from up to down, using aso-called a π pulse, the time is set such that

∫Ω(t)dt = π.

Considering |h3|>> Ω, with Ω = Ω0eiνt, with ν ∼ h3, we define another change of variables:

a0 = a0eiνt/2

a1 = a1e−iνt/2 (32)

Plugging these equations into (30), one derives for a0,1:

˙a0 = ih3 − ν

2a0 + ...

˙a1 = ih3 − ν

2a1 + ...

(33)

When ν = h3, then the drive is resonant with the atom and the a0 and a1 will rotate into each other. Sucha system could be created with a two level atom driven by an electromagnetic wave at a frequency close toits energy splitting. Such a phenomenon is called resonance. Transforming the a0 and a1 components asin (32) can be viewed as transforming under a unitary:

U = eiνt/2σz . (34)

2.5 Quantum operations

Below we define some ‘quantum operations’ which are unitary:

n = z, ωt/2 = π/2

→ U = iZ(35)

n = x, ωt/2 = π/2

→ U = iX(36)

For a Hadamard gate H:

n = x/√

2 + z/√

2, ωt/2 = π/2

U =1√2

(1 11 −1

).

(37)

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Such a unitary is called a Hadamard gate.

The Solovay-Kitaev theorem shows that with these gates as well as a π/8 rotation about Z, one can generatean arbitrary rotation on the Bloch sphere.

2.6 Tensor products

Reference: McMahon Consider a multi-partite system (also known as a composite system), with N = 2 qubits.Qubit A(B) has a Hilbert space HA(B) of dimension dA(B). The dimension of the total Hilbert space H is dAdB .An example of a state in H can be written as:

|ψAB〉 = |φA〉 ⊗ |χB〉 (38)

Where |φA〉 is in HA and |χB〉 is in HB .

Remarks:

1. If |nA〉 ∈ HA, |nB〉 ∈ HB , where |nA,B〉 is an orthonormal basis for HA,B , then |nA〉 ⊗ |nB〉 isa basis for H . The most general state can be written as a linear combination of these basis vectors:

|ψAB〉 =∑nA,nB

|nA〉 |nB〉 〈nA| 〈nB |ψAB〉 (39)

If the following is true:|ψA,B〉 6= |ψA〉 ⊗ |ψB〉 (40)

Then |ψA,B〉 is an entangled state.

2. If an operator A acts in HA and B acts in HB , then:

A⊗B |ψAB〉 = A |φA〉 ⊗B |χB〉 . (41)

3. Considering states:

|φA〉 =

(ab

)|χA〉 =

(cd

) (42)

The definition of the product state is:

|φA〉 ⊗ |χB〉 =

acadbcbd

(43)

For example, consider two qubits in |0〉A |0〉B , with the notation e.g. |01〉 ≡ |0〉A ⊗ |1〉B .

4. We make the following definitions:

|Φ±〉 =1√2

(|00〉 ± 11)

|Ψ±〉 =1√2

(|01〉 ± |10〉)(44)

These four states are called the Bell states, and they form a basis called the Bell basis.

9

2.7 Density operator

References: John Preskill’s notesSuppose that we have a quantum state of a composite system described by the following:

|ψAB〉 = a |0〉A |0〉B + b |1〉A |1〉B , (45)

such that |a|2+|b|2= 1. Suppose that one only can measure qubit A, and only cares about system A. We considerobservables MA ⊗ 1B . The expectation value is:

〈ψAB |MA ⊗ 1B |ψAB〉 = |a|2〈0|MA |0〉A + |b|2〈1|MA |1〉B = TrMAρA (46)

Where ρA = |a|2|0〉 〈0|A + |b|2|1〉 〈1|B . We call ρA the density operator for subsystem A. The physical meaningis the following: ρA is an ensemble of possible quantum states, each occuring with some probability. In this case,one example is preparing state |0〉A with probability |a|2 and state |1〉A with probability |b|2.

For the state |Φ±〉, |a|2= |b|2= 12 . In this case, 〈σx,y,z〉 = 0. One can compare this with the single qubit

state 1√2(|0〉+ |1〉), where 〈σx〉 6= 0 but 〈σy,z〉 = 0. The results obtained for a two -qubit entangled state is much

different from a single qubit state. For this reason, the density operator is described as a statistical mixture ofpure states, and can describe a more general picture of quantum states than the pure states.

We can also change the basis using a unitary operation: ρA → UρAU†. For the case of the statistical

mixture presented above (|a|2= |b|2= 1/2), we see that under any unitary transformation (in any basis), we haveρA → U 1

2U† = 1/2. Physically, this means that a maximally mixed state is a statistical mixture no matter what

basis we measure in.This is in contrast with a pure state (for example a = 1 or b = 1), in which one can always find basis with

a deterministic measurement outcome. For example, we can consider the pure state described by the densitymatrix ρ = |+x〉 〈+x|. If we measure along the Z-axis, we will of course get up and down with probability 1/2.However, along the x-axis, we will always measure +x, which is in contrast to the maximally mixed state.

Next we will look at how a state can evolve from a pure state (e.g. |0〉) into a mixed state, as described above.This clearly cannot happen as a result of unitary dynamics alone, and will thus require a new formalism. Tostart, we consider one large closed quantum system, which can be decomposed into two subsystems. In general,we can write the system in this way

|ψAB〉 =∑i,µ

ai,µ ⊗ |i〉A |µ〉B . (47)

To calculate the expectation value of some operator in subspace A, MA, this is equivalent to computing:

〈MA〉 = 〈MA ⊗ IB〉

=∑j,ν,i,µ

a∗jνaiµ 〈jA| 〈νB | MA ⊗ IB |i〉A |µ〉B

=∑i,j,µ

a∗jµaiµ 〈j|A MA |i〉A = TrAMAρA

(48)

This is because we can insert identity IA =∑k |kA〉 〈kA|, and we have

ρA =∑i,j,µ

aiµa∗jµ |i〉A 〈j|A (49)

The physical interpretation is the following. The matrix |ψ〉AB 〈ψ| is the density operator for the full systemA and B. But suppose we only care about the degrees of freedom of system A, and do not care at all aboutsubsystem B. We can simply trace over all of the degrees of freedom of subsystem B to obtain the reduced densityoperator ρA, which contains all of the information we have about subsystem A. This density operator is a muchmore general description of a system than just a pure state, since it allows us to describe subsystem A even if itis part of a larger subsystem AB. There are some important properties of this density operator:

• Hermitian: ρ†A = ρA

• ρA is positive

10

• 〈ρ2〉 ≤ 〈ρ〉

• tr ρA =∑i,µ|ai,µ|2= 1

Since the density operator has eigenvalues that are real and positive, and should sum to one, we can write it in adiagonal form ρA =

∑α pα |ψα〉 〈ψα| where

∑α pα = 1. The state described by this density matrix can be viewed

as drawn from an ensemble of different quantum states |ψα〉 which are each drawn with probability pα. Thisgives us an intuition for why ρA must be positive: in order for this density matrix to have this physical meaning,the probability pA of drawing a state |ψA〉 must be positive. In quantum mechanics, we can have amplitudesthat are negative and complex. This is what makes quantum mechanics unique, and makes quantum computerspotentially powerful. However, the underlying probabilities for measurement outcomes must always be positive.

From this result, we can make a few remarks.

• If and only if pα = 1 and all others are 0→ ρA = |ψ〉 〈ψ| is pure

• In the ensemble interpretation, |ψα〉 , pα is a mixed state as described above.

• The origin of mixed states arises from entanglement with the environment. In other words, our subsystemof interest (A) was entangled with subsystem (B, also known as the environment).

• The density matrix elements are:

〈i| ρ |j〉 =∑µ

ai,µa∗j,µ (50)

We can clearly see that the indices sum to 1:∑di=1 ρii. In addition, we see that the off diagonal elements

obey the Hermiticity condition: ρ∗ij = ρji. These density matrix elements are an important descriptionof quantum systems. the diagonal elements ρii, are known as the populations (i.e. the probabilities tomeasure the system in a particular state), and ρij are known as the coherences, which are nonzero whenthere is some well-defined phase between the states described by indices i and j.

• The density operator for a single qubit can be written as:(ρ00 ρ01

ρ10 ρ11

)(51)

For a qubit in a pure state, we can write down the density matrix immediately from the state:

ρ = (c0 |0〉+ c1 |1〉)(c∗0 〈0|+ c∗i 〈1|) (52)

and we see that the matrix elements will obey ρ10ρ01 = ρ11ρ00. We can now write the density operator asa superposition of identity and all Pauli matrices:

ρ =1

2(I+ ~P · ~σ) (53)

where

detρ =1

4(1− ~P 2) ≥ 0 (54)

because all eigenvalues are positive. It is easy to see that for |~P |= 1, it is clear that ρ describes a pure

state. For general (not necessarily pure) states, |~P |≤ 1, where |~P |= 0 corresponds to a maximally mixed

state. In fact, it turns out that ~P is a generalization of the Bloch vector, and that ~P 2 is a measure of thedegree of purity of the state in question.

2.8 Generalized evolution

We now have a generalized description of a subsystem A which is not necessarily in a pure state. Suppose initiallywe did prepare subsystem A in a pure state. How might the pure state describing subsystem A lose its purity?

11

This must occur via some dynamics involving interactions between A and the environment B. These dynamicswill be described by a Hamiltonian of the general form:

H = HA +HB +HAB , (55)

where HA(HB) is the Hamiltonian acting only on system A (B), and HAB is the interaction Hamiltonian de-scribing the interaction between system A and B.

Let us assume for simplicity that at initial t = 0, we can write down the density matrix in the form:

ρAB = ρA ⊗ |EB〉 〈EB | . (56)

At some later time t, what is ρAB? For an isolated total system AB, we can describe the system with a statevector and its dynamics with unitary evolution: |ψAB〉 → U |ψAB〉. For a density matrix ρ =

∑α pα |ψα〉 〈ψα|,

we will have evolution ρ→ UρU†.At this point, we are describing Schrodinger equation evolution to the full system AB. Now we take this

description and trace over the degrees of freedom of subsystem B. This will transform the reduced densityoperator for subsystem A:

ρA → ρ′

A = TrBρ′

AB

=∑µ

〈µ|UAB |EB〉 ρA 〈EB |U†AB |µ〉(57)

Let us define a new set of operators:Mµ = 〈µ|UAB |EB〉 (58)

which act only on system A. These are known as Kraus operators, and they act in only subsystem A. Writingthe evolution in the compact form:

ρ′

A =∑µ

MµρM†µ. (59)

This expression is known as the Kraus operator sum representation, which is the most general description of theevolution of quantum states. There are a few interesting properties to note:

• First, the operators form a completee set: ∑µ

M†µMµ = IA. (60)

Equation (60) follows from the fact that summing over all basis vectors in subsystem B yields the identityoperator:

∑µ |µ〉 〈µ| = IB

• This map is linear, Hermitian (ρ′†A = ρ

′

A), and preserves positivity. In other words, this map preserves allof the important properties of the density operator.

One can show that any linear map that preserves the trace and which is completely positive will alwayshave this operator sum representation (59). It is also possible to show that if a system with an operator sumrepresentation can be understood as dynamics of a pure state & coherent evolution on a larger, extended Hilbertspace (see Preskill’s note). Mixed states in general can be thought of as originating as a pure state on a largersystem; the most general dynamics of quantum systems can be understood as unitary evolution on a largerHilbert space. For example, subsystem A is a qubit, and subsystem B is an environment in the lab. Next, wewill discuss specific examples of such non-unitary evolution on system A.

2.9 Examples - quantum channels.

These processes describe how qubits lose their coherence.

12

2.9.1 Depolarization channel of the qubit

Suppose with probability 1− p, the state of the system is preserved, and with probability p/3 the bit flips aboutthe x axis (σx |ψ〉), with probability p/3 the phase flips (σz |ψ〉), and with probability p/3 both occur (σy |ψ〉).How can we think about these noise processes? We can use Krauss operators. In this case, they are:

M0 =√

1− p1

M1,2,3 =

√p

3σx,y,z

(61)

Such that the density operator evolves as:

ρ→ ρ′ = (1− p)ρ+∑

α=1,2,3

(p/3)σαρσα (62)

For example, a corresponding unitary evolution on the extended subspace is:

UAB |ψ〉A |0〉B =√

1− p |ψ〉A |0〉B +

√p

3

∑α

σα |ψ〉A |α〉B , (63)

where all the |α〉 are orthogonal to |0〉 as well as to each other. The environment keeps track of what the error was(the reason why this is not intuitive is because at this stage we have not specified what the interaction is betweenthe system and environment, and we have not specified anything about the environment’s degrees of freedom).Note that these states |α〉 are not necessarily unique, but in general, there exists a unitary description using alarger Hilbert space that describes decoherence of the smaller system A.

2.9.2 Dephasing channel of the qubit

Suppose with probability 1 − p the state is preserved, and with probability p there is just a phase flip (σz |ψ〉).In this case, the two Kraus operators are:

M0 =√

1− p1M1 =

√pσz

(64)

Such that:

ρ→ ρ′ = (1− p)ρ+ pσzρσz

=

(ρ00 (1− 2p)ρ01

(1− 2p)ρ10 ρ11

)(65)

When p = 1, the off diagonal components pick up a minus sign. The inherent randomness of the errors eventuallydestroys the coherence of the qubits. However, in this particular case, if p = 1, the system undergoes thedynamics where with unity probability the phase is flipped. This is coherent evolution because the phase isflipped deterministically. The coherent evolution of the larger Hilbert space for general p is:

UAB(α |0〉A + β |1〉A) |0〉B= α |0〉A (

√1− p |0〉B +

√p |1〉B) + β |1〉A (

√1− p |0〉B −

√p |1〉B)

(66)

Note if the qubit starts in state |0〉A or |1〉A, the dephasing channel has no effect. When p = 1/2, the two statesof system B multiplying the possible states of A are orthogonal to each other - the environment keeps track ofthe error, consequently, decoherence is due to the environment measuring the state of the system and that whentracing over the environment, we lose this information.

2.10 Master equation for density operator

Consider again subsystems A and B which each evolve with their own Hamiltonian and have some interactionbetween them:

H = HA +HB +HAB (67)

13

Can we find the equations of motion for the density operator for subsystem A? What is ρA? If HAB = 0, andρA(0) =

∑α pα |ψ〉A 〈ψ|A. the density operator for subsystem A evolves as:

ρA = − ih

[HA, ρA] (68)

This can be found from the Schrodinger equation. However, the case where HAB 6= 0 is much more complicated.Now the dynamics depend on the interaction HAB but it also depends on HB . To derive the equation of motionfor ρA, the so-called master equation, we first recall a necessary concept, Fermi’s Golden Rule.

2.10.1 Fermi’s golden rule

Let’s consider one closed system which has many states. We assume that there is some Hamiltonian whichcouples the initial state to every state of a continuum, but not between states of the continuum (suppose theHamiltonian is diagonal in the continuum subspace, for simplicity). Fermi’s golden rule is used to describea bound (well defined ground state) to continuum transition (many excited states |1〉k). We a wave functiondescription:

|ψ(t)〉 = c0(t) |0〉+∑k

ck(t) |k〉 (69)

The evolution equations are, from the Schrodinger equation:

c0 = −iE0

hc0 − i

∑k

〈0|H |k〉h

ck

ck = −iEkhck − i

∑k

〈k|H |0〉h

ck

(70)

First we make a transformation to eliminate diagonal elements:

cj = cje−iEjt/h (71)

Substituting into the Schrodinger equation:

˙c0 = −i∑k

〈0|H |k〉h

cke−i(Ek−E0)t/h

˙ck = −i∑k

〈k|H |0〉h

cke−i(E0−Ek)t/h

(72)

Suppose the system starts in the state |0〉 , c0 = 1. If we assume c0 = 1, and invoke perturbation theory, the firstorder correction will be:

ck =〈k|H |0〉E0 − Ek

(e−i(E0−Ek)t/h − 1) (73)

Using this description, the total probability of the system leaving the state |0〉 is:

p =∑k

|ck|2=∑k

|〈k|H |0〉 |2sin2 ((E0 − Ek)t/h)

(E0 − Ek)2(74)

In short time, the components of each sum evolves quadratically in time, since sin((E0−Ek)t/h) ≈ (E0−Ek)t/h.In the limiting case where the states |k〉 is a very broad continuum, then we must replace the sum with anintegral:

p =

∫|〈k|H |0〉 |2sin2 ((E0 − Ek)t/h)

(E0 − Ek)2ρ(Ek) dEk (75)

The ρ(Ek) here is a density of states, namely, the number of states with energy Ek. This is a sharply peakedfunction, where only the states in which E0 ∼ Ek contributes, such that the probability becomes:

p = ρ(E0)|〈k|H |0〉 |2∫ +∞

−∞

sin2(E0 − Ek)t/h

(E0 − Ek)2dEk (76)

14

Making a change of variables to (E0 − Ek)t, we find that:

p ∼ γt, (77)

where γ = πρ(E0)|〈k|H |0〉 |2. The probability of staying in zero after a short time t is p0 = 1 − γt. Theprobability leaks from state |0〉 to the continuum. Note that there is no return - the probability of staying inzero is monotonically decreasing. We can also describe these dynamics by an effective Hamiltonian evolution:

Heff = (E0 + iγ) |0〉 〈0| (78)

This imaginary term effectively describes departure of population from the initial state. We will describe this asa model for the Markovian environment. The environment is large (lots of states), dense in its spectrum (manymodes around resonance) and it is featureless. If the information leaks from the system to the environment, itimmediately disappears and it never comes back. In this case, this type of environment, a Markovian environment,is described by a Markov process that has no memory. The erasure of this memory is related to the lineardependence in time of the probability of the leakage in probability. If the dynamics are Markovian we can derivethe master equation.

2.10.2 Density operator evolution for Markovian environments

Suppose the density operator ρ(t) is given at some time t. Let us assume that the evolution is linear:

ρ(t+ δt) = ρ(t) +O(δt), (79)

where O(δt) is linear in δt. We know that the most general description of evolution of the density matrix can bewritten using the Kraus operator sum, so we can also write

ρ(t+ δt) =∑µ

MµρM†µ. (80)

We now investigate the question, what are the possible forms of Mµ?

• First, there must be one that acts like identity and are linear in δt.

M0 = I +O(δt) = O + (K − iH)δt, (81)

where K and H are Hermitian operators.

• Next, there must be operators that are proportional to√δt.

• We must preserve normalization: ∑µ

M†µMµ = I (µ 6= 0). (82)

This tells us that we must have K to be

K = −1

2

∑µ>0

L†µLµ. (83)

Now we can evaluate the differential:

ρ =ρ(t+ δt)− ρ(t)

δt= (K − iH)ρ+ ρ(K + iH) +

∑µ

LµρL†µ. (84)

We can now rewrite the terms in a more illuminating fashion. Taking the first two terms, we recognize that theterms involving H can be rewritten in terms of the commutator, and we can rewrite K using equation (83):

(K − iH)ρ+ ρ(K + iH) = −i[H, ρ]− 1

2

∑µ

(L†µLµρ+ ρL†µLµ). (85)

The resulting equation with substitution given in (85) is the most general evolution of the density operator fora Markovian environment, and is called the master equation.

Remarks:

15

1. What is the meaning of K? K plays the role of an imaginary, non-Hermitian component of the Hamiltonian.In fact, it is sometimes called the “non-Hermitian” correction to the Hamiltonian. This is the term thatresults in the decay we expect, for example from Heff given in equation (78). However, we know thatprobability should be preserved, since this is a general description of the evolution of ρ, and leaving theevolution only described by K would result in decaying total probabilities with time.

2. This brings us to Lµ, the so-called quantum jump operators. They project the density operator into certainstates, as if the environment is measuring the subsystem described by ρ. By doing so, they also preservethe normalization of the state described by ρ. If we have a pure-state ρ = |ψ〉 〈ψ|, under application of thejump operators, we have a state Lµ 〈ψ| |ψ〉L†µ, which preserves normalization.

In summary, the general description of quantum dynamics, the master equation, can be viewed as evolutionunder a non-Hermitian Hamiltonian (K − iH term), along with quantum jumps (

∑µ LµρL

†µ), which preserve

normalization.

2.10.3 Example: spontaneous emission

For example, consider an atom with two electronic states |0〉 and |1〉 (such as two orbitals of the Hydrogen atom).If the atom is in an excited state |1〉, we know the atom can emit a photon into the environment and undergo atransition |1〉A |0〉B → |0〉A |1〉B , where |ψ〉A denotes the state of the atom and |0〉B and |1〉B denotes the absenceor presence of a photon in the environment.

Note that in practice, although we are treating just two orbitals of a single atom, the total system is verycomplicated: it contains, in principle, all of the modes of the electromagnetic field in the entire universe! Ourmaster equation formalism allows us to treat just the degrees of freedom of the atom, and describe its interactionwith the environment. The jump operator L = |0〉 〈1|√γ, which describes transitions from the excited to theground state at rate γ, describes this interaction. Now using (83), we can also write down the dissipation term,K = −γ2 |1〉 〈1|. Remarkably, this is exactly the non-Hermitian term we wanted to add in the first place, such as inour un-physical Hamiltonian (78). The key here, is that the jump operator L allows us to maintain normalizationeven in the presence of this non-Hermitian evolution.

Let’s examine how these L operators by applying the quantum jump to an arbitrary density operator:

ρ→ LρL† = |0〉 ρ11 〈0| . (86)

The jump operators force the system into the state |0〉, which is what we expect when the atom undergoes aspontaneous emission event: after the jump, the system always ends in the state |0〉.

Suppose we could measure the environment, for example by putting detectors all around the atom. If at somepoint, the detector clicks, we know the photon was emitted. By recording a click, we know the atom must be inthe completely pure, normalized |0〉 state, thus purifying a potentially mixed state through measurement.

Now let us describe the evolution of the density operator under spontaneous emission. To do so, we canexamine how the density matrix elements 〈i|ρ|j〉 = ρij evolve with time. We use the master equation to arriveat the so-called optical Bloch equations:

ρ11 = −γρ11

ρ00 = γρ11

ρ01 = −γ2ρ01.

(87)

The first equation tells us how probability decays from state |1〉 due to spontaneous emission. Note however,that the second equation tells us that the population of |0〉 increases accordingly, since when population leaves|1〉 it must be preserved. The normalization condition ρ00 + ρ11 = 1 is preserved by these equations.

The final term is a little less intuitive. We can get a sense of what is happening by considering a purestate α |0〉 + β |1〉. If we now include the state of the environment, we see that spontaneous emission results inentanglement with the environment:

α(√

1− p |1〉 |0〉E +√p |0〉 |1〉E) + β |0〉 |0〉E . (88)

The coherence (or relative phase) between |0〉 and |1〉 of the atom is decaying as a result of spontaneous emission,and this is described in the master equation for ρ10. Why is the decay rate of the coherence 1

2 of that of

16

the population? Let us consider the description of the system with the non-Hermitian Hamiltonian (78). Theequation of motion for amplitude for the state |1〉 is:

c1 = −γ2c1

d

dt|c1|2 = −γ|c1|2

(89)

However, remember that this non-Hermitian Hamiltonian, does not couple to |0〉 and therefore c0 is unchanged.Considering the coherence term ρ01 = c0c

∗1, we can see this will only decay at a rate γ

2 , unlike the probability|c1|2 itself, which decays at γ. Of course, the master equation also includes the jump terms which help preservenormalization, but these do not contribute to the rate at which population is transferred between |0〉 and |1〉,since that is only described by the non-Hermitian term K.

The master equation allows for treatment of coherent and incoherent dynamics simultaneously. For example,if a quantum computer has some finite interaction with the environment, this formalism will allow us to calculate,the coherent gate errors arising from incoherent dynamics. We did not yet discuss the coherent dynamics in ourexample master equation, but one can simply employ the relevant Hamiltonian in the master equation for H,calculate the commutation relations, and obtain the result. As a shortcut however, one can just write down theSchrodinger equation (which describes only the coherent evolution), and combine this with the simple equationsgoverning the incoherent evolution given in equation (87). The two methods are equivalent, and yield the results:

ρ11 = −γρ11 + iΩρ10 − iΩ∗ρ01

ρ00 = γρ11 − iΩρ10 + iΩ∗ρ01

ρ01 = −γ2ρ01 − ihzρ01 − iΩ(ρ00 − ρ11).

(90)

Note that there is yet another equation for ρ10, but it is simply the complex conjugate of the third masterequation: ρ10 = ρ∗01. Let us examine the dynamics in general. In the case of hz = 0, the coherent evolution isa solution of sines and cosines, allowing for Rabi oscillations between state |0〉 and |1〉. However, the incoherentdynamics will result in a decay of the oscillations to some steady-state values. In fact, we can calculate the steadystate values by setting all of the time-derivatives to zero and solving the resulting system of linear equations. Thesteady state population ρ11 will generically be less than 1

2 , since spontaneous emission always favors putting usback to |0〉. Even in the case of a very strong drive, in the steady state, the population in |1〉 is at most ρ11 = 1

2 .

Remarks:

1. We can return to the intuition of a non-Hermitian Hamiltonian

Heff = H + iK = H − iγ2|1〉 〈1| (+jumps). (91)

Here, we add these quantum jumps to keep the state normalization and allow the description to be physical.The quantum jumps are especially helpful since it allows us to incorporate the physics of measuring thesystem into the dynamics. For example, we can consider performing a measurement and then doing feedbackto control the state of the system, and this is a growing tool in quantum information processing.

2. We can also consider the role of dephasing, rather than spontaneous decay. In this case, the jump operatorwill be L =

√γZ, and the non-Hermitian operator will be K = −γ2 I. In this case, note that while K looks

rather insignificant, since it is non-Hermitian, it must be treated with care as it contributes to a reductionin the purity of the state.

We have found before that this dephasing will not affect the populations, but will destroy the coherence.The purely incoherent dynamics is as follows, given by the master equation:

ρ00 = 0

ρ11 = 0

ρ01 = −2γρ01.

(92)

17

3. For a physical picture of dephasing, suppose we have a qubit which evolves under a time-dependent magneticfield in the z direction, hz(t). The phase φ in the x-y plane undergoes evolution described by precessionaround the z-axis with frequency given by hz. In this time-dependent case, we have, as seen before with achange of variables τ = hz(t)dt, that the phase between states |0〉 and |1〉 becomes at time t:

φ(t) =

∫ t

0

hz(t′)dt′. (93)

This evolution is deterministic, given an hz(t). However assume now that hz(t) is a random variable, drawnprobabilistically from some distribution such that it has a random stochastic value. The phase will nowundergo a random diffusion such that 〈φ〉 = 0, but the variance will grow such that 〈φ2〉 ∼ γt, consistentwith the master equation. Note that this is not a quantum mechanical average, but a statistical averageover the random process of phase accumulation over a random hz(t): if we make a measurement of thequbit in the x-y plane, we will no longer have a well-defined result, but rather a random result. This is whatwe mean when we say that a qubit has lost its coherence. Since the density matrix represents a statisticalaverage over each of these possible hz(t), the off-diagonal matrix elements, the coherences, go to zero ast→∞.

4. The picture defined above allows us to extend our intuition to a non-Markovian process. Now suppose thatour field hz(t) is random, but changes very slowly. For example, each experiment, we may have a randomhz(t). However, within an experiment, hz may not change, or may change very slowly. In this case, thedynamics will not be described by a master equation (recall that our derivation started with an assumptionof a Markovian environment, or a very dense continuum of states in Fermi’s Golden Rule).

While this may appear unfortunate for our understanding of the dynamics of the system, these slow, non-Markovian dynamics can often be eliminated. Suppose after some evolution time t under a random (butnearly static hz), we rotate the state of the qubit by π around the x-axis. Now, if the system evolvesfor another period t, the system will return to its original state, as if there were no stochastic hz in thefirst place. This is the idea of so-called dynamical decoupling, which is a key tool in quantum informationprocessing, also known as spin-echo in the NMR community.

2.11 Generalized measurements

Armed with a description of coherent and incoherent dynamics, we will now turn our discussion to measurements.How is the probabilistic, projective nature of measurements consistent with our quantum description thus far?How can we describe measurements in general? We will now begin to explore these questions.

As we have previously discussed, ideal projective measurements are described by projectors Pi such thatPiPj = δij , such that the density matrix is transformed to ρ → PiρPi, yielding the pure state described by Pi.The measurement occurs with probability pi = TrAPiρ.

However, we must have a description of more general measurements. In practice, states are not pure aftermost measurements in the classical world; for example, touching a table and measuring its dimensions does notturn it into a pure quantum state. Moreover, we have not yet considered the measurement device and its effecton the system, for example, photo-detectors destroy the photon that they detect. In particular, now we will usethe description of coupled system dynamics to consider a model of realistic measurements.

Suppose we want to measure system A. Suppose it couples to system B at time t = 0, and we let the twosystems evolve. Then eventually, we measure system B at time t. We will assume that we measure system Bcompletely quantum mechanically (e.g. a projective measurement). For concreteness, system B in this case coulddescribe the detector with which the measurement is made, such as a photo-detector that clicks when a photon(system A) impinges on it.

Recalling the discussion of Kraus operators, we describe the most general dynamics:

|φ〉A |0〉B → UAB |φ〉A |0〉B (94)

Inserting unity partitioned by the basis vectors of B:

|φ〉A |0〉B →∑µ

|µ〉B 〈µ|B UAB |φ〉A |0〉B

=∑µ

|µ〉BMµ |φ〉A ,(95)

18

where ere we recognize Mµ as the Kraus operator, here called ‘measurement operators’. Next, we perform aprojective measurement of system B, in which we find the system B in one of the states |µ〉B . From equation(95), this occurs with probability:

pµ = 〈φ|AM†µMµ |φ〉A , (96)

and the state of system B is projected into the state Mµ |φ〉A /√pµ. Note that these operators Mµ are not

necessarily projectors - they are Kraus operators (the projection was another operator, acting on system B).Also, the operators Mµ do not have to sum to unity (in general

∑µMµ 6= 1) and they are not necessarily

Hermitian, since they are not generally projectors.To formalize this description, we introduce another operator:

Fµ ≡M†µMµ (97)

The probability of projecting system B into state µ can now be written as pµ = TrAFµρA. These operators Fµhave the following properties:

1. Fµ are Hermitian.

2. Fµ are positive, since they describe probability outcomes.

3.∑µ F†µFµ = 1; the operators Fµ are complete.

These set of operators are called Positive Operator Value Measure (POVM). They are a formalism thatdescribes general measurements on a quantum system.

Remarks:

1. The measurement described by Fµ is very similar to conventional projective measurements, however theyare not necessarily projectors, namely, FiFj 6= δijFi in general. If one specifies the POVM Fµ, one cannotspecify the post measurement state uniquely (since that is defined by the measurement operator Mµ, andthere are multiple Mµ that yield the same Fµ).

2. Any POVM on some Hilbert space on system A can be realized as projective measurements on a largerHilbert space (in this case, the projective measurement on system A and B is equivalent to the projectivemeasurement on system B, since we assume all states |µ〉 are distinct and orthogonal). See ‘Neumark’stheorem’ in Preskill’s notes.

2.11.1 Example: POVM

In the context of distinguishing non-orthogonal states, POVMs can be utilized. Consider two parties Alice andBob, where Alice sends to Bob one two possible (non orthogonal) states: |ψ1〉 = |0〉 and |ψ2〉 = (|0〉 + |1〉)/

√2.

Bob would like to perform a measurement that determines which state she sent.

Bob constructs a detector that implements the following POVM. Consider the POVM F1 = |1〉 〈1| (√

21+√

2) and

F2 =√

21+√

212 (|0〉 − |1〉)(〈0| − 〈1|); and lastly, necessarily, F3 = 1− F1 − F2 to satisfy the completeness condition.

If Bob measures F1, he knows with certainty that |0〉 was not sent. Likewise, if Bob measures F2, then he knowsthat ψ2 was not sent. These choices would constitute a perfect set of measurements with no ambiguity if thePOVM was only F1, F2. However, there must be some ambiguity since the states are not orthogonal: Bob canalso measure F3. The third POVM F3 is important because it guarantees that the operators sum to unity, andallows for some ambiguity in trying to distinguish non-orthogonal states.

Remarks:

1. The post-measurement state is not defined by POVM. For example, consider the weak measurement POVM:

F0 = |0〉 〈0|+ (1− ε) |1〉 〈1|F1 = ε |1〉 〈1|

(98)

If ε→ 0, then F0 becomes unity - the measurement only weakly perturbs the system. Naively, one may saythat the post-measurement state is proportional to Fµ |φ〉A. But, in our original description, the operatoracting on the state is the Kraus operator Mµ, not necessarily Fµ. However, the Mµ are not unique for a

particular Fµ! For example, M0 = |0〉 〈0|+ eiφ√

1− ε |1〉 〈1| satisfies F0 = M†0M0 for any φ.

19

2. One way to interpret these effects is in terms of open quantum systems and decoherence & depolarization.In this case, the environment measures the system and learns something about it. If one could keep trackof all degrees of freedom of the environment, then we could learn exactly the information that it gainsabout the system. In practice the environment contains many degrees of freedom, and we must trace overthem (add all outcomes). That is precisely the motivation for the operator sum representation and Krausevolution: entanglement, measurement and decoherence are all very closely connected.

2.12 Entanglement

We will now move on from measurements and generalized dynamics, and formalize our definition of entanglement,which plays a role in all of the concepts we have discussed thus far. We define a state as entangled when:

|ψ〉AB 6= |ψ〉A ⊗ |ψ〉B (99)

This is a description of real-world systems, and as we have seen, it can be thought of as an ‘enemy’ of purity whendescribing system-environment interactions and measurements, but it is also a resource for quantum information.In the following sections, we will discuss how entanglement is a resource.

2.12.1 Example: Bell states

After preparing the Bell state |φ+〉AB = |00〉+|11〉√2

, and measuring the state of each qubit in the computational

basis, the outcomes of the measurements will be correlated. This correlation is sometimes called non-localcorrelation, because the qubits A and B can be physically separated from each other. This can be used forsuperdense coding - by sending one qubit from one location to another, two classical bits can be sent.

2.12.2 Example: multiple qubits

Consider multiple qubits and quantum function evaluation which consists of two registers:

|01....10〉 |00...1〉 (100)

Where the first register is called x and the second is the function is the function evaluation f(x). Consider theunitary that evaluates f(x):

U∑i

|xi〉 |0...0〉 →∑i

|xi〉 |f(xi)〉 (101)

For N qubits, there are 2N combinations of basis states. To specify the state of this type, one requires 2N

complex numbers! This becomes intractable very quickly for a classical computer and is the basis of quantumsimulation and quantum supremacy.

2.13 Properties of Bell states

The Bell states are special two-qubit entangled states, which are employed in many applications of quantumentanglement. They are described by the expressions:

|Φ±〉 =1√2

( |00〉 ± |11〉 )

|Ψ±〉 =1√2

( |01〉 ± |10〉 ).(102)

The notations of using |Φ±〉 and |Ψ±〉 are quite widely used by the community: the Φ or Ψ describes whetherthe qubits are in the same or different states, respectively, and the ± describes whether or not there is a π phasebetween the two terms. Important properties about the Bell states include:

1. The Bell states form an orthogonal basis in HA ⊗HB .

2. They translate into one another via local operations, which are operations acting only on one of the qubits.For example, consider XA |Φ+〉. The X operation flips the first qubit on both states |00〉 and |11〉 in thesuperposition |Φ+〉, so we clearly see that XA |Φ+〉 = |Ψ+〉.

20

3. The expectation value of any single-particle observable is zero. Mathematically, ∀σA,Bα , 〈σA,Bα 〉 = 0.

4. The Bell states are simultaneous eigenstates of two-qubit operators XAXB , YAYB , ZAZB . We can see this,for example, by considering XAXB |Φ+〉. Each Xi flips its respective bit, leaving us with XAXB |Φ+〉 =|Φ+〉. This is in contrast to the case of a single qubit, where [X,Y ] = 0 ensures that a single qubit cannotbe a simultaneous eigenstate of X and Y . However, note that [XAXB , YAYB ] = 0. This will have somevery important consequences, for example, in the Bell inequalities as we will learn later on. While singlequbit expectation values are all zero, the two-qubit expectation values 〈σA,iσB,i〉 are in general nonzero.

5. Consider a partial measurement, a measurement on just one of the qubits. For example, consider a mea-surement of only qubit A in the Z basis. The possible outcomes are as follows:

0→ |0A〉 〈0A|Φ+〉 → |0A〉 |0B〉1→ |1A〉 〈1A|Φ+〉 → |1A〉 |1B〉 .

(103)

Now consider also a measurement of the same state in the X basis, where we have the possible outcomes:

+→ |+XA〉 〈+XA |ΦAB+〉 = |+XA〉 (|0B〉+ |1B〉) = |+XA〉 |+XB 〉− → |−XA〉 〈−XA |ΦAB+〉 = |−XA〉 (|0B〉 − |1B〉) = |−XA〉 |−XB 〉 .

(104)

If we measure one of the qubits, we will always get a well-defined outcome on the second state. Thisoutcome will be determined by (1) the basis of the measurement and (2) the outcome of the measurement.The Bell states have some non-classical correlation where the correlations between qubits extends beyonda single basis and can be maintained simultaneously in multiple non-commuting bases.

6. If we trace over one of the qubits, we get a completely mixed state. Mathematically, the reduced densitymatrix is described by:

ρA = TrB |ΨAB〉 〈ΨAB | =1

21. (105)

This has important consequences for classifying entangled states. If we are given an arbitrary state, we canask (1) are the qubits entangled? And (2) how entangled are they? This could be for a general entangledstate in a large Hilbert space, but we can get a good intuition from the two qubit case, where we see thattracing over one of the qubits in the maximally entangled Bell-states results in a maximally mixed state.This is powerful, because given an arbitrary quantum state, it is not always obvious whether or not it isentangled. For example, consider the two states:

|ψ〉 =1

2( |00〉+ |01〉+ |10〉+ |11〉 )

|ψ〉 =1

2( |00〉+ |01〉+ |10〉 − |11〉 ).

(106)

These states seem similar, but the first state is a simple product (unentangled) state |+x+x〉, whereas thesecond state is a maximally entangled state. In this case, examining at the reduced density matrix couldhelp us identify a lack of entanglement in the first state, and the presence of entanglement in the second.

2.14 Schmidt Decomposition

The Schmidt decomposition is a tool used to quantify and test for entanglement within a two-qubit pure state.The Schmidt decomposition can be formulated as a theorem as follows:

Theorem: ∀ pure state |ψAB〉 can be written as

|ψAB〉 =∑i

√Mi |ui〉A |vi〉B , (107)

where |ui〉, |vi〉 are an orthonormal basis in HA and HB . Mi ≥ 0 are known as the Schmidt coefficients.Proof: The reduced density operator for system A is ρA = TrB |ψ〉AB 〈ψ|. We choose the basis |nA〉 for

subsystem A such that ρA is diagonal. We can decompose the state as:

|ψAB〉 =∑n,µ

an,µ |nA〉 |µB〉 ≡∑n

|nA〉 ⊗ |nB〉 (108)

21

where we have defined |nB〉 ≡∑µ an,µ |µB〉, which denotes the change of basis for subsystem B for choosing the

basis where ρA is diagonal. Now the reduced density matrix is:

ρA = TrB∑n,m

|nA〉 |nB〉 〈mA| 〈mB | =∑n,m

|nA〉 〈mA| |〈mB |nB〉 |2. (109)

Recall we chose the basis |nA〉 such that ρA was diagonal, so it follows that 〈mB |nB〉 is proportional to δm,n,and specifically 〈mB |nB〉 ≡Mnδm,n. From this, (107) follows.

Examining for example the state:

|ψ〉 =1

2( |00〉+ |01〉+ |10〉+ |11〉 ), (110)

it is clear this is not in its Schmidt form given by equation (107). Rather, the description of the state as|ψ〉 = |+A,x〉 |+B,x〉 is the Schmidt form, in which it is clearly not entangled.

Remarks:

1. Consider the concept of purification of mixed states. Any density operator ρA =∑iMi |ui〉 〈ui| can always

be represented as a pure state |ψAB〉 in some extended Hilbert space given by (107). This is intimatelyconnected to what we have studied previously: generalized evolution, and POVMs can always be understoodin terms of unitary evolution in a larger Hilbert space.

2. Schmidt decomposition eq (107), tells us whether or not a state is an entangled state of two subsystems.In particular, |ψAB〉 is separable if and only if one Mi = 1 and all other Mj = 0. We can also see this interms of the reduced density operator ρA: if ρA it is a pure state, then the state is separable.

On the other hand, we can consider the opposite limit: the state is maximally entangled if all Mi = 1d where

d = min(dimHA,dimHB). We can therefore define a degree of entanglement as the purity of the reduceddensity operator. For example, if we want to quantify the degree of entanglement of two qubits, we cansimply trace over one of the qubits and calculate the length of the remaining Bloch vector to understandthe degree of entanglement.

3. Entanglement entropy, also known as the von Neumann entropy is defined as:

S( |ψAB〉 ) ≡ S(ρA) = −TrρAlog2ρA = −∑x

Mxlog2Mx. (111)

This quantity is the analogue of Shannon entropy in classical information theory. For example, if the state|ψAB〉 is pure, we will have S(|ψAB〉) = 0 since only one Mi = 1 and the rest are zero. For a maximallyentangled state, we will have S(|ψAB〉) = log2d.

This metric is very useful in theory for classifying entanglement in large systems. However, it is generallynot as useful experimentally. One reason is because the term log2ρ is challenging to measure. There is afamily of these entropy measures, for example Reyni entropy, which is proportional to ρ2, which can bemeasured in some experimental systems. However, the main challenge is that we assume explicitly thatwe started with a pure state of the two subsystems |ψAB〉. In practice, no system is completely isolatedfrom the environment, so it is very challenging to start in a pure state. Then, when we measure an impurereduced density matrix ρA, it is very challenging to ensure that any mixedness or impurity, arises fromentanglement with subsystem B (generally desirable) rather than with the rest of the environment (whichis undesirable).

2.15 Unitary evolution of entangled states

Now that we have defined entanglement more formally, we will discuss how entangled states evolve with time, orthe dynamics of entangled states.

Suppose we have two systems which evolve under a Hamiltonian H = HA + HB . In this case, the unitaryevolution can be written as a product of unitary evolution on the individual subsystems: U = UA⊗UB . Can suchevolution change the degree of entanglement? Examining the Schmidt decomposition, eq (107) shows that thisunitary evolution will not change individual Schmidt numbers, but simply transform the underlying basis vectors

22

|uj〉A |vj〉B : in order to change the degree of entanglement, we will need some component of our HamiltonianHAB which acts on both components of our state. Practically, there must be an interaction between the qubitsthat governs and evolves their joint quantum state |ψAB〉 as a combination of qubits.

We can further examine this evolution by returning to the states described by eqs (106). To change thefirst state into the second, the sign of the |11〉 component only must be changed. This will correspond to theHamiltonian:

HAB = g |1〉A 〈1| ⊗ |1〉B 〈1|

≡ g

2(1A − ZA)(1B − ZB).

(112)

This Hamiltonian is proportional to ZAZB , which is a product of the Pauli operators on each system A and B.Such a Hamiltonian, HAB , is required in order to change the degree entanglement the two-qubit system, sinceany other Hamiltonians, HA and HB , correspond to unitary operations on individual subsystems A and B. ThisHamiltonian maps the basis states as follows:

|00〉 → |00〉|01〉 → |11〉|10〉 → |10〉|11〉 → e−iφt |11〉

(113)

We can implement the desired − sign on the |11〉 component by choosing the time gt = π, which result in theso-called controlled-phase gate, or controlled-Z rotation. We can see logically that this operation will apply a πphase flip on the second (target) qubit conditional on the state of the first (control) qubit.

One interesting property of the controlled-phase unitary is that the control and target qubits and the sameresult still occurs. This is a unique property of the controlled-phase gate, and is why we draw the diagram forthis controlled-phase gate in a symmetric form.

Another important two-qubit gate is a controlled-NOT (CNOT) gate. This is a specific implementation ofthe broader class of the controlled-unitary gates. In these gates, conditioned on the state of the control qubit, weimplement a unitary rotation U on the target qubit. For the CNOT gate, U = X, so we invert the target qubitstate, conditional on the state of the first qubit (i.e., only if the first qubit is in state |1〉). The CNOT gate hasthe following mapping:

|00〉 → |00〉|01〉 → |10〉|10〉 → |11〉|11〉 → |10〉 .

(114)

One can show that the CNOT operation can be obtained from the C-phase gate using two Hadamard transforma-tions: CNOT ≡ HBCZHB . One important limitation on these multi-qubit unitaries is the no-cloning theorem.This states that if we have a system in the state |φA〉 |0B〉, where |φA〉 is an arbitrary quantum state, thereexists no unitary UAB that clones the state |φA〉 into the second register: there is no such UAB that satisfiesUAB |φA〉 |0B〉 = |φA〉 |φB〉 for general states |φ〉. The no-cloning theorem can be proved by taking an innerproduct of two instances:

〈ψA|φA〉 = 〈ψAU†AB |UABφA〉 = 〈ψA|φA〉 〈ψB |φB〉 , (115)

And thus |φ〉 and |ψ〉 cannot be general states. This is the basis for the field of quantum cryptography (quantumkey distribution). It is interesting to note that if cloning did in fact exist, one could actually use entanglementto communicate faster than the speed of light, as explored on the homework.

2.16 Information content of Bell States

.As we have seen, single qubit measurements of Bell states always yield expectation values of zero. That begs

the question, what is the information content is there in Bell states? There are actually two bits of information:

1. Parity : the parity is measured in the ZZ basis and distinguishes whether the Bell state is in the Φ orΨ manifold.

23

2. the phase is measured by XX and determines if the state is in the + or - manifold.

To convert from Bell states to classical computational basis states, and extract these two bits of information,we need to eliminate the entanglement which requires two-qubit unitaries UAB . These are not always easy toimplement physically - for example, the two qubits could be spatially separated. In this lecture, we will be givenan entangled state and want to learn something about it just by measuring single qubit operators. The focus ofthe following sections will be to understand how we can leverage the correlations between qubits and entangledstates of physically separated qubits.

Suppose Alice and Bob are spatially separated and share a singlet state |Ψ−〉. Alice measures her qubit in theZ basis with the operator σAZ with measurement operators MA

0 = |0〉 〈0|A and MA1 = |1〉 〈1|A. She measures each

with 50% occurrence rates, by definition of the Bell states. The value that Alice measures determines Bob’s statesince they share a Bell pair. In particular, the following Kraus operators act in Bob’s subspace: MB

0 = |0〉 〈0|Band MB

1 = |1〉 〈1|B .Let us consider Bob’s qubit throughout the process thus far. Before Alice performs a measurement, what is

the density matrix of Bob’s qubit? He has the identity as his reduced density matrix, because they share a Bellpair at first. If Alice then measures her qubit and sends this information to Bob, what state is Bob’s qubit in?Bob has a pure state given by Alice’s outcome. Conceptually, this is interesting - a measurement in Alice’s labinfluences Bob’s qubit, even though they are spatially separated. Einstein grappled with this concept over 60years ago when he formulated his Principle of Locality.

2.16.1 Einstein’s Principle of Locality & Hidden Variable Theory

.Einstein postulated that a complete description of physical reality requires that an action on subsystem A

must not instantaneously affect the description of B if they are spatially separated. His conclusion was that thequantum mechanical description of the wave function must not be complete: we cannot have a measurement ofA affect the description of the state of B. This is also referred to as the EPR paradox.

To resolve the paradox, the most general way is to employ a local hidden variable λ. If we consider adescription of the Bloch vector in a particular direction n, and measure it in a direction m, we know that weobtain the outgoing state |m〉 with probability p0 = cos2 θ and the antiparallel state |−m〉 with probability1 − cos2 θ. The role of λ is to provide the measurement outcome - λ is a variable between 0 and 1, and in thiscase is equal to p0 for m and 1− p0 for −m. In this description, by only examining a single qubit, we cannot testor gain information about every degree of freedom at play, since our measurements is incomplete as λ is hidden,by definition.

2.16.2 Bell inequalities

.Another resolution to the EPR paradox uses the so-called Bell inequalities. Consider the state |Ψ−〉. The

measurement outcomes of Alice and Bob will be perfectly anti-correlated, and likewise the phase measurementswill be anti-correlated, if we measure in any basis. If Alice and Bob perform measurements in arbitrary baseswith respect to each other, will the results be correlated, will they not be correlated at all or will they dependon the basis? The Bell inequalities quantify the answer to that question.

Suppose Alice measures along a direction a with projection operators MA0 = 1

2 (1+a·σ) and MA1 = 1

2 (1−a·σ),

and Bob measures along a direction b with projection operators MB0 = 1

2 (1 + b · σ) and MB1 = 1

2 (1 − b · σ).Since, given the state |Ψ−〉AB , the results are completely anti-correlated, we can replace σB with −σA, such

that: MB0 = 1

2 (1+ b · σA) and MB1 = 1

2 (1− b · σA). The possible outcomes of the measurement are 00, 01, 10, 11for Alice and Bob’s measurements, no matter what basis they measure in (these are the eigenvalues of the Paulimatrices). The probabilities of these outcomes according to quantum mechanics is:

P00(a, b) = 〈Ψ−| 12

(1+ a · σA)(1− b · σA) |Ψ−〉

=1

4(1 + 〈a · σA〉 − 〈b · σA〉 − 〈a · σAb · σA〉

= Tr(ρAa · σA)− Tr(ρB a · σA)− 1

4(1− cos θ) =

1

4(1− cos θ) = P11(a, b)

(116)

24

Where θ is the angle between a and b. If Alice and Bob measure along the same direction, they always have anti-correlated results, and if they measure in opposite directions, they will obtain correlated results. Additionally,Alice and Bob can make measurements with some small angle between their measurement basis, and still get somedegree of correlation between them. Suppose Alice measured in one basis, and Bob measures in another. If Bobsends his result to Alice, she knows what her measurement result would have been if she had measured in Bob’sbasis, because of the perfect correlation between qubits. This might be surprising - to further understand thecorrelations between measurements, we turn to a classical picture, and compare it to the quantum one describedabove.

We can quantify a figure of merit S corresponding to measurements along three axis such that:

S = Psame(eA1 ,−eB2 ) + Psame(e

A1 ,−eB3 ) + Psame(e

A2 ,−eB3 ) ≥ 1. (117)

Classically, if the probabilities are already determined and given, at least one of these terms must be one, sincethere are only two options of the measurement outcomes, but there are three independent measurements.

To understand the output quantum mechanically, let’s define z = e1, and e2 and e3 in a plane, with angle2π/3 between the vectors. We can examine the probabilities:

Psame(eA1 ,−eB2 ) = Psame(e

A1 ,−eB3 ) = Psame(e

A2 ,−eB3 ) = 1/4 (118)

Since cos(2π/3) = −1/2. The value of S = 3/4 is less than 1 - a violation of the classical result. By using theprinciples of quantum mechanics to calculate what correlations of the measured values we expect, we were ableto get a result which violates the case where measurements are predetermined, and thus we have disproved localhidden variable theory.

This formulation of the Bell in equality is not unique, it is a particular example utilizing the singlet state.There are many different ways to construct the Bell inequalities, and the formulation depends on the specificstate that one starts with and the vectors that are measured.

Remarks.

1. Correlations predicted by the quantum theory are incompatible with local hidden variable theory.

2. There is a larger class of Bell inequalities. For example, the |Φ+〉 state should be able to generate non-classical correlations (correlated measurements along any axis). The general version is the CHSH inequality.Four directions a1,2 and b1,2 are chosen, and Alice and Bob calculate the quantity C = |〈a1b1〉+ 〈a2b1〉 −〈a1b2〉+ 〈a2b2〉|≤ 2.

2.16.3 Violation of Bell’s inequalities

.Why are we studying this in a quantum information course? What can we learn from Bell’s inequality?

1. Bell’s inequalities are an entanglement witness. If we measure a violation of Bell’s inequality, it means thatwe must have an entangled state. For two qubits in a pure state, this might be trivially expected, butfor a larger system where the degree of entanglement is less clear, the Bell inequalities become useful. Inaddition, for mixed states of two qubits, we can use Bell’s inequalities to check if two qubits are entangled,as well as characterize it.

2. If a density matrix fails the violate Bell’s inequality, we cannot draw the conclusion that the state is notentangled, namely, there are entangled states that do not violate Bell’s inequalities. For example, the purestate |Ψ〉 =

√1− ε |01〉 −

√ε |10〉 does not violate a Bell inequality for small ε.

3. Maximal violation of Bell’s inequalities occurs with the Bell states; the minimum S = 3/4 and the maximumC = 2

√2 occur with the Bell states. The values of S and C change monotonically with the degree of

entanglement, so they are a tool to characterize how entangled a state is which is experimentally feasible.

25

2.16.4 Loopholes

.Throughout the literature, there have been various experimental attempts to measure Bell inequalities (with

a recently successful measurement, see 2015 Hensen et. al, Nature). In 1985, Apsect performed an experimentwith Cs decay. A Cs atom has a ground and excited state, and can decay by emitting a pair of photons, onehorizontally and one vertically polarized. The state of the two photons can be written as a bell state, in particular,

|Ψ+〉 = |V H〉+|HV 〉√2

. Aspect measured a Bell inequality and found it to violate the classical result. However, the

field was not satisfied with this experiment result because there were some important experimental loopholes:

1. Fair sampling : in the experiment, many the results are not measured - photons in the experiment getabsorbed by something that is not the detector, or are deflected before reaching the detector. Not detectinga significant fraction of photons could allow for a hidden variable to control which photons were eliminated.This was resolved in 2004 using trapped ions with better readout.

2. Locality is a more challenging loophole. Alice and Bob must be space-like separated states to show a trueviolation of Bell inequalities. If Alice’s measurement requires a finite amount of time, such that Bob iswithin Alice’s lightcone, then a violation of Bell’s inequality would be consistent with Einstein’s theory andwould not result in a paradox. This was resolved in 2015 in Delft (Hensen et. al, Nature). Clearly, thisfield is a frontier of modern research – the current technical challenges involve the questions, how far apartcan we separate entangled states and how quickly can we measure them?

2.16.5 Applications of Bell’s inequalities to quantum information processing

Device-independent quantum communication is sometimes called quantum key distribution (QKD). In this sec-tion, we will explore QKD further. Suppose Alice and Bob are spatially separated and share a singlet state |Ψ−〉.They perform measurements in randomly chosen basis as described by the CHSH inequality. Alice and Bob thenpublicly share the basis they performed their measurements in. The values for which the basis was the sameconstitutes a secret key. However, if the basis was at some angle with respect to one another, they can use thosemeasurements to instead check the CHSH inequality (measure C). If C ≥ 2, they share a genuinely entangledstate, and there must have not been an eavesdropper measuring the key (which would destroy the entanglement).In other words, if one can prove C ≥ 2, one can verify that the channel is secure. This is a device-independenttechnique, because it can be applied to any channel, independent of the detector and channel hardware.

2.17 Applications of entanglement

In the previous lecture, we discussed the EPR paradox, the Bell inequalities, and its application to quantuminformation processing. Now that we have shown that quantum mechanics is indeed complete, as well as touchedon some of the applications of entanglement, we will discuss further what entanglement can be used for as aresource. The applications and impact of entanglement include the following:

1. Quantum key distribution is verifiably secure. As a review, in the Ekert protocol, Alice and Bob create anentangled pair, then measure in the z or x basis, and subsequently publicly announce the basis. If the basisis the same, then they generate a bit in their secret key. If the basis is different, they can use those bits tocheck the Bell inequality and confirm that the channel is secure.

2. In superdense coding, Alice and Bob share a Bell pair, for example |Φ+〉AB . Alice applies four operatorsto her qubit, 1, X, Z,XZ. Depending on the operation, the state will change to a different Bell state. Forexample, applying Z yields the state |Φ−〉AB . After encoding these four possibilities, she sends her qubitto Bob. If Bob makes a measurement in the Bell basis, he can decode one of these four outcomes, which istwo classical bits.

3. In quantum teleportation, Bell pairs can be used to communicate quantum bits over a classical channel.Suppose Alice has a state |φ1〉 = α |0〉+ β |1〉. Suppose she wants to send this qubit to Bob but only overa classical channel. The two parties have several options: option 1 is that Alice could measure her qubitand send Bob the result. Bob could then reconstruct the state to the best of his ability. It’s clear that thisreconstruction will not be perfect - in fact, the fidelity of reconstruction will be 2/3, as seen on a previoushomework problem. However, there is option 2: if Alice and Bob share an entangled pair, she can send this

26

Figure 2: A circuit representation of quantum teleportation. Picture from Wikipedia.

state to Bob perfectly. Suppose that Alice has qubit 2 and Bob holds qubit 3, and they share a Bell pairin the state |Ψ−〉23. In that case, we can rewrite the three qubit state as:

|ψ〉123 = |φ1〉 |Ψ−23〉 =α√2

(|001〉 − |010〉+β√2

(|101〉 − |110〉)

=1

2

(|Ψ−12〉 (−α |0〉3 − β |1〉3) + |Ψ+

12〉 (−α |0〉3 + β |1〉3) + |Φ−12〉 (α |1〉3 + β |0〉3) + |Φ+12〉 (α |1〉3 − β |0〉3)

)(119)

Rewriting the three-qubit state way suggests a strategy for transmitting the state |φ1〉. Suppose now thatAlice performs a Bell basis measurement of her qubits (1 & 2). At this point, she will obtain one of fourdifferent outcomes and transmit the classical result to Bob. Then, when Bob learns which state she mea-sures, he can apply a unitary transformation to his qubit to reconstruct the original state. For example,if Alice transmits to Bob that she measures the singlet state, Bob will do nothing. If Alice transmits thatshe measures |Ψ+〉, then Bob needs to apply the Z operator to recover the qubit |φ2〉. To summarize theprotocol, Alice makes a Bell basis measurement of her two qubits, sends Bob the result, and Bob applies aunitary operation to reconstruct the initial state.

Remarks about quantum teleportation:

(a) This result is consistent with the no cloning theorem because the state |φ1〉 must be destroyed to sendit to Bob.

(b) Quantum teleportation does not violate causality. There is no instantaneous propagation of infor-mation, since Alice needs to transmit her measurement result for Bob to properly reconstruct thequbit.

(c) The quantum circuit for teleportation can be represented as shown in figure 2.

The first part of the circuit, which rotates between the product state basis and the Bell state basis,allows for the Bell state measurement between Alice’s qubits. That is the CNOT and Hadamardgate. By performing a measurement of qubits 1, 2, two classical bits of information are extracted andconditional on that result, a single qubit unitary is applied to Bob’s qubit.

2.18 Mixed state entanglement

.When we considered entangled states in quantum teleportation and the violation of Bell inequalities, we

discussed the situation where Alice and Bob can share a Bell pair, and we assumed that the two party systemwas isolated, such that the Bell pair was a pure state. In general, there will be other degrees of freedom, forexample in an environment C, which will interact with qubits A and B. In this case, the state of Alice and Bob’squbits is no longer pure and we must describe it with a density matrix.

Specifically, we have a two-qubit density matrix ρAB =∑pα |ψα〉AB 〈ψα|AB . Let us define uncorrelated states

as the joint density operator ρAB = ρA⊗ρB . Secondly, separable states are defined as ρsep =∑k pkρ

kA⊗ρkB , such

that∑k pk = 1. The separable state can be prepared if Alice generates a random number k which is a sample

from the distribution pk, and sends this random number to Bob, whereby Bob performs some unitary rotationthat depends on the number. In this case, Alice and Bob have a-priori agreement in that Alice prepares ρkA andBob prepares his state in ρkB . By using local operations and a classical communication channel, Alice and Bob

27

can prepare such separable states. The state is correlated, but classically correlated, or more technically, thestate ρsep can be prepared by local operations and classical communication (LOCC). The intuition here is thatBell state have non local quantum correlations, so we cannot prepare a Bell state using LOCC. In light of thesedefinitions, we will define entangled states as ρAB 6=

∑k pk(ρkA + ρkB).

To consider this definition, we describe some quantities in further detail. A state of of two qubits can bewritten as:

ρAB =

3∑i,j=0

ρij |i〉AB 〈j|AB , (120)

whereas, a Pauli representation of a two-qubit entangled state is more complicated but still manageable: Ai =1AB , XA1B , YA1B ... is a set of 16 operators that define a basis for the two-qubit density matrix. We couldexpress a single qubit in terms of 4 operators, similarly, we can describe a two qubit state in terms of 16 operators.For example, consider the density operator:

ρ = f |Φ+〉 〈Φ+|+ (1− f) |Φ−〉 〈Φ−| . (121)

This is a statistical mixture of two states, one |Φ+〉 and one |Φ−〉. Is this state entangled for arbitrary f? If so,how much entanglement is there? Suppose that f = 1/2. In this case, there is equal probability of |Φ+〉 and|Φ−〉. The density operator will be:

ρ =1

2|Φ+〉 〈Φ+|+ 1

2|Φ−〉 〈Φ−| = 1

2|00〉 〈00|+ 1

2|11〉 〈11| . (122)

This matrix only has terms on the diagonal and is a statistical mixture of |00〉 and |11〉, two separable states,thus, it is not entangled. A signature of entanglement is non-zero off diagonal matrix elements in the 2 qubitdensity matrix, all of which in this case are zero. More specifically, the part of the density matrix responsiblefor entanglement are non-zero off diagonal terms in which both qubits change state between the ket and the brapart, for example, |00〉 〈11|, and |01〉 〈10|. These are akin to the coherences of the single-qubit density matrix.We will call these multi-qubit coherences, which can signify the presence of entanglement. (Note however, thatjust because a density matrix may have nonzero multi-qubit coherences, that does not mean it’s entangled - forexample, the pure and separable state |+〉 |+〉 has nonzero two-qubit coherences). For the example above inequation (121), we see that in the case of f = 1

2 , these terms vanish.

Remarks about ρAB :

1. We can always define a reduced density operator ρA = TrBρAB , and this reduced density operator cansometimes help us to check if the qubits are entangled. However, if ρAB were not in a pure state tobegin with (as is the case with mixed-state entanglement), this reduced density operator is not helpful inquantifying the entanglement between subsystems A and B. For example, consider the case ρAB = ρA⊗ρBwhere ρA = 1

21 (completely mixed). In this case, even though the reduced density matrix ρA is completelymixed, we cannot conclude that it is entangled with subsystem B (and in fact it is not)! Only if we cancertify that ρAB was pure to begin with can we use the reduced density operator purity as an entanglementwitness.

2. Positive Partial Transposition (PPT) Suppose we have two subsystems, A and B with bases |m〉 and|µ〉 in HA and HB . We can write the density matrix elements as:

ρmµ,nν = 〈m| 〈µ|ρ|n〉 |ν〉 . (123)

Partially transposing this density matrix (exchanging the indices µ and ν), we obtain the matrix ρᵀBAB =ρmν,nµ. For separable states, this corresponds to only transposing the reduced density operator for subsys-tem B:

ρᵀBAB =∑p

ρ(1)A ⊗ ρ

(1)ᵀB . (124)

Since ρ(i)ᵀB is non-negative (based on the positivity of the density matrix ρ

(i)B ), ρᵀBAB will be nonnegative.

The PPT criteria says exactly this: if ρAB is separable, then ρᵀBAB is nonnegative. For 2 qubits, PPT isa necessary and sufficient condition for entanglement. Note that this is not true for higher dimensionalsystems (e.g. qutrits).

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3. Entanglement witnesses. An entanglement witness forms a hyperplane in a large Hilbert space, guaranteeingthat everything outside this hyperplane is entangled. For an entanglement witness operator W, if ρ isseparable, then 〈W 〉 = TrρW > 0. In other words, if we measure 〈W 〉 < 0, then ρ is entangled.

One prominent example is called entanglement fidelity. Suppose we attempt to prepare the Bell state|ψ〉 = |Φ+〉, but instead only prepare some mixed state ρ. We define the fidelity as:

F ≡ 〈ψ|ρ|ψ〉 . (125)

In the case where |ψ〉 is one of the Bell states, showing that F > 12 proves that the state is entangled. We

can consider, for example, the state described by (121). We find that in the example of (121):

F+ = 〈Φ+|ρAB |Φ+〉 = f

F− = 〈Φ−|ρAB |Φ−〉 = 1− f,(126)

as expected, the state is entangled as long as f > 12 or (1− f) > 1

2 , meaning that it is only not necessarilyentangled if f = 1

2 exactly. However, for example we might consider instead the fidelity associated with|Ψ−〉:

〈Ψ−|ρ|Ψ−〉 = 0, (127)

but this does not mean that the state described by (121) is not entangled! We can construct witnesses asdescribed above (in this case the witness would be of the form):

W = − |Φ+〉 〈Φ+|+ 1

2(128)

such that 〈W 〉 < 0 for f > 12 ). However, there exist many of these witnesses, and if any of them has an

expectation value less than 0, then we can conclude our state is entangled. The key is to choose anentanglement witness appropriate for the mixed state at hand.

Another example which we have already seen that uses entanglement witness is the violation of a Bellinequality (one just needs to appropriately modify the inequality to have the form 〈W 〉 < 0). We mustcarefully note that there are states which are entangled, such that there exists a witness that provesentanglement, but do not violate any Bell inequality.

4. The Werner states are described by a statistical mixture of a Bell state, with the fully mixed state. Forexample:

ρ = (1− p) |Φ+〉 〈Φ+|+ p1

41. (129)

One might think that naively the portion of the density operator p 141 does not contribute at all to an

entanglement fidelity with the state |Φ+〉. However, there is some contribution to the fidelity even fromthe component of the density matrix that is fully mixed, since we can write 1 = |Φ+〉 〈Φ+|+ |Φ−〉 〈Φ−|+|Ψ+〉 〈Ψ+| + |Ψ−〉 〈Ψ−|. In general, the Werner states are useful since they describe a statistical mixtureof getting a particular entangled state some fraction 1− p of the time, and some completely random statesome other fraction p of the time, which is true in many experiments.

5. Similar to our consideration of the Kraus operator evolution of the single qubit reduced density matrix, wecan also examine similar generalized dynamics for subsystem AB:

ρ→ ρ′ =∑µ

MABµ ρMAB†

µ +∑µ′

MAµ′ρM

A†µ′ +

∑µ′′

MBµ′′ρM

B†µ′′ . (130)

Here, we have separated the Kraus operator sum into terms that act on subsystem AB jointly, as wellas terms that act only on subsystems A and B. These individual terms can describe individual coherentrotations of single qubits, but could also describe how the individual qubits interact with the environmentand decohere (independent of the other qubits).

As an example, if we consider Markovian dephasing of individual qubits, we will see that our masterequation will have the form

ρ = −i[H, ρ] + LA(ρ) + LB(ρ), (131)

29

where Li(ρ) describes the non-unitary evolution of subsystem i. We expect that the dynamics should beadditive in the case where the qubits interact with the environment and dephase independently. This willgive rise to density matrix evolution of the form

ρ00,11 = −(γA + γB)ρ00,11 + ..., (132)

such that in the example of an initial Bell state ∝ |00〉 + |11〉 undergoing dephasing, with single qubitdephasing rates given by γA (γB), the state undergoes a random flip ZA OR ZB , at rate γA + γB , eitherof which leaves us in the opposite Bell state ∝ |00〉 − |11〉. As such, the two qubit coherences decay at rateγA + γB .

As a final note, there are also decoherence processes which act collectively on multiple qubits at once, andthese cannot be described by the individual terms acting on just subsystems A or B in (130). These areinteresting, because they can in some cases result in entanglement between subsystems A and B. However,they are also potentially a major problem, since all modern theories of quantum error correction assumeexplicitly that errors are uncorrelated and system evolution can be described in a form such as the examplein (132).

2.19 Multipartite entanglement

We will now consider the properties of multipartite entanglement amongst arbitrary numbers of qubits. First,let us consider how we might generate such an entangled state, starting with a pure initial state and using theoperations we have already discussed. We remember that we can produce a Bell state with a circuit consistingof a Hadamard rotation followed by a CNOT gate. However, if we add additional qubits and additional CNOTgates, we can produce entangled states of multiple qubits. For two qubits, recall that the Hadamard and CNOTgate produces the state:

|00〉+ |11〉√2

. (133)

It is clear that repeating this process for N qubits we will obtain the N-partite entangled state:

|0〉⊗N + |1〉⊗N√2

. (134)

2.19.1 GHZ states

This specific type of state in equation (134) is called the |GHZ〉N state. It is a Schrodinger cat-like state wherethe system is in a superposition of (sometimes many-qubit) different states |00..0〉 and |11..1〉. Some propertiesof these states include:

1. They are eigenstates of the product operators Z1Z213...1N , 11Z2Z314...1N , ... 11...ZN−1ZN , as well asX1X2...XN with eigenvalue +1. These operators which return the state to itself are often called stabilizersand are an important concept in quantum error correction.

2. Consider the case of N = 3. Using the fact that ZX = iY , one can show that the GHZ state is also aneigenstate of operators Y1Y2X3, X1Y2Y3, Y1X2Y3 with eigenvalues −1. If we now consider the generalizationof hidden variable theory to three qubits, we will find that 〈X1X2X3〉 = −1, which we know is not thecase. In fact, 〉X1X2X3〈= 1 as noted above. This is the basis for the so-called GHZM paradox, which isyet another violation of local realism and a generalization of Bell’s theorem.

3. Partial measurements on |GHZ〉 states: consider a measurement of the first qubit in the computational

basis. We will always measure a result 0 or 1, which will collapse the full state into either |0〉⊗N or |1〉⊗N .Consider now a measurement of the first qubit in the |+x〉 |−x〉 basis. The amplitude of the GHZ statein |±x〉1 is:

( 〈0|1 ± 〈1|1 ) |GHZN 〉√2

, (135)

30

and the GHZ state is projected into the states:

|0〉⊗N−1+ |1〉⊗N−1

√2

|0〉⊗N−1 − |1〉⊗N−1

√2

(136)

By choosing the X basis to measure the first qubit, we don’t reveal information about whether the remainingqubits are in |0〉 or |1〉. This is also a consequence of the fact that the initial GHZ state is an eigenstate ofX1X2...XN , such that the measurement of X1 leaves the state in an eigenstate of X2...XN with eigenvaluedetermined by the measurement outcome.

4. We can analyze the effect of decoherence on |GHZ〉. If we trace over even just one of the qubits, we obtainthe state:

Tr1 |GHZ〉 〈GHZ| = 1

2( |00..〉 〈00..|+ |11..〉 〈11..| ). (137)

This is a mixed state with no entanglement between qubits - there are no off-diagonal elements of thisdensity matrix. This shows that GHZ states are very sensitive - decoherence of even a single qubit com-pletely destroys the entanglement, which experimentally presents a major challenge in quantum informationprocessing.

5. Quantum metrology, however, takes advantage of this sensitivity to precisely sense the environment. Forexample, if a magnetic field couples to a qubit through a Hamiltonian H = hZ, and we would like tomeasure the field h, we can use GHZ states for enhanced sensing. To see this, consider a single qubit

prepared in the |0〉+|1〉√2

, which will acquire a relative phase φ(t) = 2ht under precession of the field h. With

a single measurement, we can distinguish between relative phases of φ = 0 and φ = π (the two orthogonalstates |±x〉, defining a smallest possible detectable change in φ, a sensitivity of δφ ∼ 1. We can repeatthis procedure with N independent spins (or repeat this with a single qubit N times), which will give animprovement in our phase sensitivity that scales as δφ ∼ 1√

N.

However, if we start with the GHZ state, under evolution under the Hamiltonian H it evolves into thestate:

|0〉⊗N + eiφN |1〉⊗N√2

. (138)

To detect the phase change in a single measurement, we get δφN ∼ π, in other words, δφ ∼ 1N , corre-

sponding to the two orthogonal states where Nφ = 0 and Nφ = π. This is known as Heisenberg-limitedsensitivity, which is known to be the best possible scaling of the sensitivity with particle number, and isnot possible without entanglement.

6. The fidelity F = 〈ΨGHZ | ρ |ΨGHZ〉 is an entanglement witness for the GHZ state. The GHZ state isunique in that an entanglement witness can be defined, and it can also be measured - as it only dependson 4 elements of the density matrix (for most general N entangled states, we need to measure 2N variouselements of the density matrix). A fidelity F > 1/2 is sufficient for entanglement.

2.19.2 W states

Another example of a multi-partite entangled state is the so-called W state:

|ΨW 〉 =1√N

(|100...0〉+ |010...0〉+ |001...0〉+ ...) (139)

For example for N = 2, we recover the Bell state |Ψ+〉, which can be converted into the GHZ type Bell state|Φ+〉 with local operations. However, for N > 2, |ΨW 〉 cannot be converted with local operations into a GHZtype state. Consider the situation where we trace over one of the qubits (e.g. qubit 1). Then:

Tr1 |ΨW 〉 〈ΨW | =1

N|000...0〉 〈000...0|+ N − 1

N|ΨW,N−1〉 〈ΨW,N−1| (140)

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Figure 3: A circuit representation of the Cluster state.

This state is still mostly entangled, since the contribution from the first term is small. Unlike the GHZ state, ifwe trace over a single qubit, there is a significant amount of entanglement preserved, which makes these statesmore robust to noise. Additionally, the W state can be considered as a collective excitation in the many qubitsystem.

2.19.3 Cluster states

One last example of a multiparticle entangled state is the cluster state. We can introduce it with a circuit, asshown in figure 3. This state can be written as:

|ΨC〉 =1

2N/2ΠNa=1(|0〉a + Za+1 |1a〉) (141)

Where we let ZN+1 = 1. A powerful way to represent cluster states is with a graph. If each vertex is a qubitprepared in |+x〉 and each qubit is connected by an edge which corresponds to a CZ gate, then the qubitscomprise a chain. One useful property of the cluster state is that it is an eigenstate of the operators S1 = X1Z2,Sj = Zj−1XjZj+1, ... SN = ZN−1XN . The eigenvalues are all +1, such that these operators are stabilizers forthe cluster state.

2.19.4 Remarks about multipartitle entanglement

1. Graph states can be generalized: for example, a central qubit connected to other qubits is equivalent to aGHZ state up to local operations. Graph theory can be used to see which states are entangled.

2. A 2D cluster states can be a very powerful resource. It allows for universal quantum computation just bymeasurements. These 2D cluster states contain enough entanglement to perform any quantum manipulationpossible, provided that the state is large enough for the computation at hand. In fact, quantum supremacyand quantum computers can be defined in terms of their ability to create a 2D cluster state. Proofs ofquantum supremacy are often based on a reduction to creating a 2D cluster state.

3. Degree of entanglement

Another way to quantify entanglement in a many-body system is to divide the qubits into two subsectionsand trace over one of the subsections. For example, tracing over half of the qubits (system B) in the GHZstate gives the density matrix:

ρA = TrB |ΨGHZ〉 〈ΨGHZ | = |0..0〉N/2 〈0..0|+ |1..1〉N/2 〈1..1| (142)

Calculating the entanglement entropy, we find S(ρA) = 1, which is small compared to the maximal entropyfor N qubits of N/2. Despite the fact that the GHZ state is undeniably entangled, the degree of entan-glement in terms of the entropy is actually quite small. Similar calculations for W and cluster states yieldsimilar results - entanglement entropy on the order of 1 or a constant, as opposed to linear in N . Situationsin which the entanglement entropy grows with the number of particles is sometimes called the volume lawof the entanglement entropy: the entanglement grows as a function of the system size. Systems in whichthe entanglement entropy is constant is described as an area law entanglement. Intuitively, for area lawentanglement, there exists a boundary such that in making a cut in the graph (tracing over the degrees of

32

freedom on one side), one breaks a single entangled pair bond in the graph, as opposed to order N entangledpairs. Note that making a cut for the 2D cluster state gives an entanglement entropy of

√N . As expected,

if subsystem A and B are all to all entangled, then there will be ∼ N bonds broken when tracing over halfthe qubits. In fact, because of the exponential scaling of the Hilbert space with the number of qubits, moststates in the Hilbert space obey volume law entanglement.

4. In general, describing and quantifying entanglement is hard. This is the goal of quantum simulation, andquantum state tomography. These goals require many calculations and measurements, proportional to anexponential of the number of qubits N .

5. Do we need to study the zoo of entangled states that are volume law entangled? To describe the mostgeneral states, we require 2N complex amplitudes. However, most desired states that we encounter inquantum information processing are separable, which have only ∼ N complex numbers. This questionmotivates the search for special classes of entangled states which is broader than separable states, but doesnot include everything, called physically accessible states. Is there a generic description for states like this?This subject is an area of current research and is relevant for quantum supremacy and quantum simulation.The class of states are sometimes called Multiparticle Entangled Renormalization Ansatz (MERA) states,and are created with a circuit. Another approach to this is called tensor networks, associated with the classof tensor network states.

2.19.5 Tensor network states

Tensor networks are a tool to represent a subset of many-body entangled states in an insightful way. Specifically,to start, represent a qubit as a pair of d-dimensional systems - extend the qubits’ Hilbert space to d dimensions,then entangle these extra degrees of freedom in so-called bonds. Each bond, at site e, represents a maximallyentangled state |Ie,e+1〉 ∼

∑di=1 |αe〉 |αe+1〉.

A 1D tensor network state is called a Matrix Product State (MPS). An MPS is a map on each cite whichtakes the pairs of the auxiliary indices and maps them into qubit states:

Pe =

2∑i=1

D∑α,β=1

Aei,α,β |ie〉 〈α, βe| , (143)

such that the state is:

|ΨMPS〉 = ΠNe=1Pe |I12〉 |I23〉 ... |IN,1〉 =

2∑i1,...iN

TrA1i1 ...A

NiN |i1...iN 〉 . (144)

The state is specified by N dxd dimensional matrices Asij . Suppose we introduce a cut and calculate the

entanglement entropy. In this MPS case, specifically, we only cut one bond, so the maximal degree of entropycreated is proportional to the dimension of the bond. This allows for a construction of a class of entangled statewith a specific degree of entanglement, and allows for systematically keeping track of the degree of entanglementintroduced. For example, GHZ states, W states, cluster states all have bond dimension equal to 2. These MPSstates are useful both for analytics and numerics of studying many body entangled quantum systems.

3 Quantum algorithms

We have discussed some applications of quantum states in cryptography, metrology and communication. Itis interesting to ask if these ideas can accelerate solving computational problems. In classical computers, theinformation is represented in a binary encoding which can represent numbers from 0 to 2n−1, where n is thenumber of qubits. All of the operations of classical computers can be composed as a set of universal operationsor universal gates. One set is NOT, AND, OR, and COPY. Conversely, in quantum computation, the dynamicsare unitary and therefore reversible, and in general they have probabilistic outcomes. We will find that theseproperties can be leveraged to construct specific algorithms that have an advantage over classical computers, andwe will work to quantify the advantage.

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3.1 Simple quantum algorithms

We have managed to cover a wide foundational base of material. Coronavirus, SARS, communism, capitalism,republicans, democrats will come and go, but this understanding will stay. In this section, we will discuss somecanonical examples of simple quantum algorithms.

3.1.1 Quantum parallelism example

Let’s design a quantum unitary which implements the function:

(x1, ..., xn)→ (x1, ...xn, f(x)). (145)

First, we define the unitary Uf on n input qubits to be:

Uf |x〉n |y〉m = |x〉n |y ⊕ f(x)〉m , (146)

where ⊕ is modulo 2 bitwise addition. Note that Uf |x〉n |0〉 = |x〉n |f(x)〉, that Uf is unitary, and U−1f = Uf . By

preparing a superposition of all the input states, and applying the unitary, we will evolve all of the superpositions.Also note that H⊗n |0〉n prepares the combination of all possible bitstrings, which we can express as H⊗n |0〉n =

12n/2

∑0<x<2n |x〉n. Exploiting the idea of quantum parallelism, we apply the unitary to this superposition of

states:

UfH⊗n |0〉n |0〉m =

1

2n/2

∑n

|x〉n |f(xn)〉m (147)

Although we have been able to evolve all of the inputs at once, in part showing how quantum computers canbe powerful, the measurement and its outcome is probabilistic. If we measure the output, we always find justone value at random. In practice, there is no way to find all f(xn) without 2n repetitions. To overcome this,we would like to use additional operations to find some relationship between the f(xn) withoutactually revealing the values of f(xn). The art of quantum algorithms is to use additional operations to findrelationships between different f(xn) without revealing their values. Let us explore some examples.

3.1.2 Deutsch’s problem

If we have a function f : 0, 1 → 0, 1 and want to find if the function is constant or not, we can use a quantumcomputer to reveal the answer in a single measurement. Classically, there is no choice except to evaluate thefunction for different x and determine if the answer is constant. Quantum mechanically, we can evaluate thisquestion in a single function evaluation. The quantum algorithm evaluated on state |x〉 is as follows:

Uf |x〉1 (|0〉2 − |1〉2) = |x〉1 (|f(x)〉2 − |1⊕ f(x)〉2) = |x〉1 (−1)f(x)(|0〉2 − |1〉2) (148)

Now, preparing the input state in a superposition, we find:

Uf (|0〉1 + |1〉1)(|0〉2 − |1〉2) = ((−1)f(0) |0〉1 + (−1)f(1) |1〉1)(|0〉2 − |1〉2) (149)

By subsequently measuring the first qubit in the X basis, we can evaluate if f(0) = f(1) or not, namely, if thefunction f is constant.

Now we will consider the corresponding quantum circuits, which is generally a convenient and intuitive wayto think about quantum algorithms. There are only 4 possible functions that are 1 to 1 that can be created:

1. x→ 0

2. x→ x

3. x→ x⊕ 1

4. x→ 1

The circuit representation that implements all of those functions are (see figure 4):

1. Do nothing on both qubits

2. Control X with target on second qubit

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Figure 4: A circuit representation the possible functions in Deutsch’s problem. Figure from Mermin.

Figure 5: Various circuit simplifications. Figure from Mermin.

3. X on second qubit and control X with target as second qubit

4. X on second qubit

Inserting Hadamard gates before and after the unitary on both qubits, we can exploit quantum parallelism (figure6). We start with the first qubit in |0〉 and the second qubit in |1〉. Various circuit simplifications can be made(see figure 5) to confirm that when f is constant, the output is 0, and if it is balanced the output is 1.

Now we can generalize this procedure to a slightly more complex case.

3.1.3 Deutsch-Jozsa algorithm

Suppose we have a function f(xn), of n bits. We are told that the function is drawn from the sample of twopossible functions: f(xn) = c constant for all xn, and balanced, which means f(xn) = 0 for 50% of the cases,and 1 for the other 50% of cases. In one function evaluation, we can find out whether the function is constant orbalanced. We again evaluate the function on a superposition of states:

UfH⊗n |0〉n (|0〉 − |1〉)/

√2 =

1

2n/2

∑x

(−1)f(x) |x〉 (|0〉 − |1〉)/√

2 (150)

35

Figure 6: A circuit representation the possible functions in Deutsch’s problem, including the Hadamard gatesand corresponding circuit simplifications. Figure from Mermin.

We again apply Hadamard gates on each of the qubits:

H⊗nUfH⊗n |0〉n (|0〉 − |1〉)/

√2 = H⊗n

1

2n/2

∑x

(−1)f(x) |x〉 (|0〉 − |1〉)/√

2. (151)

Suppose f(x) is constant. Then (−1)f(x) is a constant factor we can take out of the sum, and the H appliedto all the qubits would return the zero qubit back on all the n registers, such that |0〉n → |0〉n. For a constantfunction, with probability 1 we will get the same qubits returned. Similarly, when the function is balanced, theprobability to get |0〉n is zero.

A mathematical remark: The Hadamard gate on a single qubit can be written as H |x〉1 = |0〉+ (−1)x |1〉 =∑1y=0(−1)x1·y1 |y1〉. Evaluating the Hadamard on many qubits, we have:

H⊗n |x〉n =1

2n/2

1∑yn−1=0

...

1∑y0=0

(−1)∑n−1j=0 xjyj |yn−1...y0〉 =

1

2n/2

2n−1∑y=0

(−1)x·y |y〉n . (152)

Here x · y = x0y0 ⊕ x1y1 ⊕ ... is a sum of products of corresponding bits, modulo 2. The state in equation (151)can also be written as:

|ψ〉 =1

2n

∑y

∑x

(−1)x·y+f(x) |y〉 (|0〉 − |1〉)/√

2. (153)

Again if f(x) = c, then the state is:

|ψ〉 =1

2n(−1)cΠn

j=1(

x∑xj=0

(−1)yjxj ) |y〉 (|0〉 − |1〉)/√

2 = δy,0 |y〉 (|0〉 − |1〉)/√

2, (154)

such that the amplitude of the y = 0 term will be 1, as expected. If f(x) is balanced, we have∑x(−1)f(x) = 0

so the amplitude of the |y〉 = |0〉 term is zero.How does this compare to classical algorithms? Worst case, for a guaranteed answer, we would have to

evaluate half of the values, plus one queries, to make sure that it is not balanced. Quantum mechanically, asshown, we can use a single function evaluation.

3.1.4 Bertstein-Vazirani problem

Another example of a quantum algorithm is the following. Consider the function f(x) = a · x, with the goal tofind a. Classically, if we want to isolate the mth bit of a, we need to multiply a to the number 2m assuming ais written in binary, where here 2m = (0, 0, ...1m, ...0). If we want to learn n bits, we can apply f to n values of

36

Figure 7: A circuit representation of the Bernstein Vazirani problem. Figure from Mermin.

Figure 8: A circuit simplification for the Bernstein Vazirani problem. After this simplification, the result becomesmore clear: the state of the input register is changed to a. Figure from Mermin.

x = 2m where 0 ≤ m ≤ n. Quantum mechanically, we can do it in one step. The reason is similar to what wehave discussed previously. Again using the same concepts and circuit, we end with the state in eq. (153), wherehere f(x) = a · x. Performing the sum over x:

∑x

(−1)x·y+a·x = Πnj=1

1∑xj=0

(−1)(yj+ai)xi = δy,a (155)

We will receive all digits of a in a single measurement, since all bits will destructively interfere unless y = a.We can evaluate this in a circuit, as shown in figures 7 and 8. We can again think of this function f(x) as anoracle.

3.1.5 Summary: simple quantum algorithms

We have now discussed several examples of toy quantum algorithms that illustrate how, in principle, quantuminformation can speed up computations.

• If we consider a function that takes some binary string x1...xn → (x1...xn, f(x)), we defined a so-calledoracle Uf which acts in the following way:

Uf |x〉n |y〉m = |x〉n |y ⊕ f(x)〉m , (156)

where ⊕ denotes addition modulo 2.

• We constructed a superposition of all input states by applying

H⊗n |0〉1 ... |0〉n =1

2n/2

∑x<2n

|x〉n . (157)

37

• Just applying this transformation and measuring does not give rise to any sort of speedup, since in eachmeasurement we only get the functional value for one of the inputs. Instead, we considered using theOracle to get a final state |ψf 〉 = H⊗nUf , which gave us the speedup in determining whether the functionis constant or balanced.

• By speedup, we mean that the Deutsch algorithm requires 2 function evaluations classically, and only 1quantum mechanically. On the other hand the Deutsch-Jozsa algorithm (for an n-bit function), we needonly 1 quantum step, as opposed to ∼ 2n (actually, polynomial in n to be more precise/generous to classicalalgorithms). In the Bernstein-Vazirani algorithm, as opposed to n steps, we only needed 1.

3.2 Simon’s algorithm

Suppose we have a two-to-one function of n bits f(x1...xn) where one value of this function corresponds to twoinput bit strings. This function reduces n → n − 1 bits. We consider the case that f(x) = f(y) if and only ifx = y⊕ a, or x⊕ y = a, such that f(x⊕ a) = y. This function is periodic with period a, and the goal of Simon’sproblem is to find this period.

Let us consider how many function evaluations it would take to solve this problem. In the classical case, wewould need to feed the values x1, x2, x3 (where here these are ordinary numbers, not binary digits), and find i, jsuch that f(xi) = f(xj) and a = xi ⊕ xj . Suppose we evaluate m different values of x, and we did not find thisequality. Then we know that a 6= xi ⊕ xj for the m values evaluated. By evaluating m inputs, we tested atmost 1

2m(m − 1) values of a. In total, there are 2n possible values of a. This means we need m ∼ 2n/2 calls ofthe classical function in order to find a on average! This shows (roughly) that the classical complexity of thisproblem is exponential.

Now, let us consider the quantum approach to solving this problem. As usual, we consider application of theoracle to a superposition of all inputs, yielding:

Uf1

2n/2

∑x=0

|x〉 |0〉 =1

2n/2

∑|x〉 |f(x)〉 . (158)

Next, we will measure the function register, which gives an output to 1 out of 2n−1 possible values. Recall thatthere are two values of x that return the same f(x). This measurement will therefore collapse the data registerto the state which is a superposition of those two values

1√2

( |x0〉+ |x0 ⊕ a〉 ) (159)

where x0 is the value for which f(x0) agrees with the measurement result for f .This seems like an incredible development, since we immediately create a superposition of two states separated

by the period of interest. In practice, however, this is of limited utility. Of course, we can measure the dataregister and we will get either x0 or x0 ⊕ a at random. If we repeat this procedure, remember that we will get adifferent random x0, so we will not actually get any speedup (we get exponentially unlucky, it turns out).

The solution, again, is not to measure this at random. We instead apply the Hadamard transform to theentire data register, to obtain:

|ψout〉 =1√2H⊗n( |x0〉+ |x0 ⊕ a〉 )

=1

2(n+1)/2

∑((−1)x0·y + (−1)(x0⊕a)·y) |y〉

(160)

Physically, this is a transformation back to the original basis, with phase factors now given by the functionevaluations. Note that we can rewrite:

(−1)(x0⊕a).y = (−1)x0·y(−1)a·y. (161)

If a · y = 1, then the coefficient in front of the corresponding state |y〉 goes to 0. Therefore, we are left with:

|ψout〉 =1

2(n−1)/2

∑a·y=0

(−1)x0·y |y〉 . (162)

38

Now, once we measure any |y〉, suppose we measure |yj〉, we immediately know that a · yj = 0 for the yj valuereceived, and we can make the statement that:

n−1∑i=0

yji ai = 0 mod2. (163)

Simon’s algorithm then repeats this process O(n) times, such that with high probability, we get n linearindependent equations for bits ai, which we can use to reconstruct the number a ≡ a0a1...an. A few remarks:

1. One can show that with n + x repetitions, the success probability becomes ps > 1 − 12x+1 (see Mermin

appendix for the derivation).

2. We have shown a quantum algorithm which accomplishes a certain task in O(n) whereas the best knownclassical algorithm required 2n/2, getting a a true exponential speedup. Of course, this is again a bit of acontrived problem since it requires knowing the oracle (which in turn, requires knowledge of a in the firstplace).

3.3 Quantum search algorithm

We will now consider Grover’s search algorithm, another oracle-based algorithm, but one that can now potentiallybe made useful. Suppose we have a large, unsorted database with N items. Our goal is to search for one specificitem, labeled w.

To formulate this in terms of the oracle, we can consider a function f(x) where x ∈ 0, ..., N − 1, wheref(x = w) = 1 and f(x 6= w) = 0. Here, the problem is to find w. Classically, we must apply f(x) to all valuesof x until it returns 1. On average, this requires N/2 function evaluations to find the solution (ps = 1/2). In theworst case, if we are particularly unlucky, we will need to check N − 1 elements before success (ps = 1).

In the quantum approach, we again define the quantum oracle:

Uf |x〉 |y〉 → |x〉 |y ⊕ f(x)〉 (164)

where |x〉 is the data register, and |y〉 is the function evaluation register. In this case, we start with the functionregister in the superposition state |y〉 = 1√

2(|0〉 − |1〉). The quantum oracle yields (for this particular y):

Uf |x〉|0〉 − |1〉√

2= (−1)f(x) |x〉 |0〉 − |1〉√

2. (165)

Since the oracle applies identity to the function register, we have that:

Uf |x〉 = (−1)f(x) |x〉 . (166)

The oracle changes the phase for x = w, so we must have the oracle:

Uf = 1− 2 |w〉 〈w| . (167)

Even though we can express the oracle in this way, we don’t necessarily know this w. Mathematically, Uf is ahuge matrix, and if we think of the oracle as a black box, it is nontrivial to find this w.

However, in the following we will show that by using the oracle, we will only need√N evaluations of the

function to find w with probability close to 1. This is the essence of Grover’s search algorithm. In summary, wedo not know the value of w, a vector in H , and the goal is to find it. The steps, in summary, of the GroverSearch Algorithm are:

1. Prepare the superposition state |s〉 = 1√N

∑N−1x=0 |x〉

2. Construct a unitary Us = 2 |s〉 〈s| − 1. This corresponds to a reflection of any vector in H perpendicularto |s〉, about the vector |s〉 (the sign will be flipped with respect to the direction |s〉).

39

3. Apply the Grover iteration:R = UsUf (168)

Where the matrix R is given by:Rm |s〉 ≈ |w〉 , (169)

and where m ∼√N . The relationship between |s〉 and |w〉 is simply that they have nonzero overlap 1/

√N .

The state |s〉 does not favor in any way the state |w〉 - it is simply required to have some small nonzerooverlap. Let’s consider the Grover iteration acting on state |w〉:

UsUf |w〉 = (1−2 |s〉 〈s|)(1−2 |w〉 〈w|) |w〉 = |w〉−2 |s〉 〈s|w〉 = 2 |s〉 〈s|)(1−2 |w〉 〈w|) |w〉 = |w〉−2 |s〉 /√N.

(170)

To simplify the expression, we use another vector |r〉 ≡ 1√N−1

∑x 6=w |x〉 =

√NN−1 |s〉 −

1√N−1

|w〉, such

that:

UsUf |w〉 = |w〉 − 2

N|w〉 − 2

√N − 1

N|r〉 = cos θ |w〉 − sin θ |r〉 . (171)

Moreover, we can express θ in terms of N : we have θ = 1− 2N or θ ∼ 2/

√N for large N , and UsUf |w〉 =

sin θ |w〉+ cos θ |r〉.We note that UsUf operates within the |w〉 , |r〉 subspace only:

UsUf

(|w〉|r〉

)=

(cos θ − sin θsin θ cos θ

)(|w〉|r〉

). (172)

We can interpret the entire evolution geometrically. The vector |s〉 is an angle in the |r〉, |w〉 plane, bysome particular angle θ/2 from axis |r〉 to |w〉. Then, Uf rotates |s〉 counterclockwise by angle θ. Lastly,Us flips this vector about the original vector |s〉, such that UsUf |s〉 is a vector in the |r〉, |w〉 plane, byangle θ from |r〉 to |w〉. Therefore, the vector |s〉 has been rotated in total by angle θ/2 towards |w〉 in this2D space.

Applying the Grover iteration k times:

(UsUf )k =

(cos kθ − sin kθsin kθ cos kθ

)(173)

For an initial state |s〉 ∼ |r〉, then after k rotations, such that kθ = π/2, then (UsUf )k |s〉 ≈ |w〉. The

number of steps will be k = π2

1θ = π

4

√N , such that the Grover algorithm requires ∼

√N iterations to find

the element |w〉. Recall that the classical result requires N , so there is a polynomial speedup achieved withGrover’s search algorithm using a quantum computer.

Remarks:

1. With the geometric expression of the algorithm, it is clear that the root of the speed up is from interference.The interference in the N-dimensional H space is between amplitudes (e.g, 〈w|s〉), not intensities (e.g,|〈w|s〉 |2): the amplitude of the overlap of the two states |r〉 and |w〉 is changed as the vector rotates witheach Grover iteration. When the amplitudes add, the intensity squares, hence the quadratic speedup.Additionally, because of the algorithm’s reliance on intererence, we can view this quantum computation asan interferometer - we do not even need to consider qubits for this process! Considering classical waves, oreven many states of a single atom can both exhibit the same interference that is required to execute thesearch properly. In fact, when this algorithm was first introduced, there were multiple experimental groupstaking this approach. However, one needs to be extremely careful about scaling of other resources: the costof other resources that are needed, such as the measurement, can negate the speedup. For example, usingoptics (and the interference of light waves), N beamsplitters are needed, and when N is large, the relativequadratic speedup is a moot point.

2. However, the search algorithm can be implemented efficiently using qubits and using a quantum circuit.Specifically, if we have entries which iterate from x ∈ 0, ..., N − 1, we can encode them using n = log2Nqubits. The steps are as follows:

• Prepare initial state |s〉 = H⊗n |0〉 ... |0〉

40

• We can express Us = 2 |s〉 〈s|−1 as H⊗n(2 |00...0〉 〈00...0|−1)H⊗n. The quantity (2 |00...0〉 〈00...0|−1)provides a phase of -1 for all states, except for |00...0〉, for which the phase is unity. To implementthis term, we define an n qubit controlled phase rotation CnZ as:

•••••

Z

Adding X gates on any qubit allows for conditioning for a phase flip if the qubit is in |0〉. For example,if the oracle uses the state |w〉 = |10010〉, then the circuit would be as follows:

x4 •x3 X • X

x2 X • X

x1 •x0 X • X

y X y ⊕ f(x)

3. How do we implement an n qubit controlled gate using only two-qubit gates? For n = 2, we have alreadyCZ and CNOT gates, etc. Now, consider n = 3. A controlled unitary squared gate (where M = U2) is:

••

M

This gate can be implemented as a sequence of two-qubit operations as follows:

• • •• •

U U† U

(one can see this by iterating through the different states of the inputs of the top two qubits, and confirmingthat U2 is applied only if these are in state |11〉). If we want to implement for example C2Z, then weconstruct the unitary such that U2 = Z, namely a π/2 phase, and apply the conditional operations as givenin the figure above. Another gate that we have already seen in the homework of this type is the Toffoligate. Converting many-qubit controlled gates to two-qubit gates is polynomial overhead. This is generallynot a problem, but for near-term quantum devices, it can be quite challenging. The idea of co-design is todevelop hardware informed by the particular algorithm that one wants to execute.

4. Suppose we would like to implement the gate CnZ where n is the number of the control qubits and Z isthe phase flip. To do this using only two-qubit or three-qubit gates, we can introduce n− 2 ancilla helperqubits. See, for example, the case of n = 5, where three ancillary qubits are used to implement a C5Z gateon the 6 qubit register below:

|0〉 •

|0〉 • •

|0〉 • •• •• •• •• •••

41

In general, this can be performed with 2(n − 2)C2X gates, or ∼ 10n CNOT gates. Using this qubitimplementation, Grover’s search can be implemented with ∼ C

√N logN steps, where here logN is the

cost of implementing multi-qubit gates and allocating ancillary qubits (recall that the number of qubits isn = logN).

5. Grover’s algorithm is optimal: the√N speedup is the best possible speedup. (See Preskill’s notes for the

proof).

6. Grover’s search algorithm is probably not actually useful for a database search, because it requires aspecial kind of database where the oracle is already known, and provided to us. However, the algorithmis important because it solves a problem without any internal structure. All of the previous problems hadinternal structure (e.g. periodicity) which we could reveal by operating on the qubits and performingcollective measurements. The power of these quantum algorithms is extended by Grover’s search becauseit is applied to a problem where there is no particular requisite structure. For this reason, Grover’s searchcan be applied to many different kind of problems: many complex optimization problems (including someNP complete ones) can be cast as a search problem, with ∼

√N speedup.

3.4 Quantum Fourier Transform

The Quantum Fourier Transform (QFT, not to be confused with quantum field theory) is an example of analgorithm that is immediately useful, unlike the previous examples. The QFT is similar to Simon’s algorithm,in that it finds a if f(x ⊕ a) = f(x). It is motivated by the idea that the problem of period finding has somestructure, and we can use quantum interference to reveal that structure. Recall that the key operation of Simon’salgorithm was performing Hadamard gates on all qubits:

H⊗n |x〉 =

N∑y=0

(−1)x·y |y〉 (174)

where we used multiplication modulo two bitwise, x · y.The question we would like to study now is can quantum computers find a period in conventional

algebra? Suppose for example we have a function that is periodic:

f(x) = f(x+mr), (175)

where m is an arbitrary integer. The goal is to find the period r for such a function. For intuition on interference,it is helpful to turn to optics. Consider impinging a beam of light on a surface. If we have a rough surface andyou send a beam of light towards it, the “grating” modulates the phase of the reflected light,giving it an effectivemomentum kick, leading to a reflection at a different angle. The light will constructively interfere only at specificpoints, depending on the periodicity of the grooves of grating, which determines the angle corresponding toconstructive interference precisely. This is similar to the kind of interference that allows us to find the period ofa function.

3.4.1 Discrete Fourier transform

First, we review the discrete Fourier transform. Suppose we have a certain function of integers f(x). The discreteFourier transform is

f(x)→ g(y) =

N−1∑x=0

e2πixy/Nf(x). (176)

Here, the function f(x) is defined on the interval [0, ..., N − 1], such that x and y are integers. Note that inthis case xy is a conventional product. If the f(x) is a 1-1 function, then the discrete Fourier transform is anNxN matrix that takes f(x)→ g(y). Classically, calculating the Fourier transform requires N2 operations (onefor each element of the matrix). This can be sped up using the ‘Fast Fourier Transform,’ which calculates thediscrete Fourier transform in N logN operations.

42

3.4.2 Quantum Fourier Transform definition

The definition of the QFT is analogous to the discrete Fourier transform:

QFT |x〉 → 1√N

∑e2πixy/N |y〉 . (177)

To implement this unitary transformation, we will make use of the n qubits to represent the N vectors in theHilbert space efficiently, such that N = 2n as previously stated. However, we must first clarify some notation.

Mathematical remark

We will use the binary representation, such that x = x020 + x121 + ..., xn−12n−1 ≡ (xn−1, ..., x0). The bi-nary fraction is defined as 0.xn−1....x0 ≡ xn−1

2n−1 + xn−2

2n−2 + ..., x0

20 . The reason why this representation is useful canbe understood as follows: we need to write the phase factor in front of each |y〉 for arbitrary integers, and we canuse similar algebra as Simon’s problem if the conventional multiplicative number xy is considered in this binaryrepresentation. Consider the phase Φ = 2πxy

2n = 2π2n (y020 + ..., yn−12n−1)(x020 + ..., xn−12n−1). Let’s further

examine these terms:

Φ = 2πyn−1(x0

2+ x1 + 2x2 + ...) + 2πyn−2(

x0

4+x1

2+ x2 + ...) + ... (178)

All of the terms that give rise to 2π phase shifts are irrelevant (for example, the ones proportional to x1 and 2x2

in the first sum, recalling that each xi, yi is 0 or 1). The sums above are reminiscent of the binary fraction:

Φ = 2π(yn−10.x0 + yn−20.x1x0 + yn−30.x2x1x0 + ...) + ... (179)

It becomes evident that the phase in the discrete Fourier transform can be written as a product of x and ywritten as binary representation. With this mathematical remark understood, we can write the QFT as:

QFT |xn−1...x0〉 =1

2n/2(|0〉+ e2πi0.x0 |1〉)⊗ (|0〉+ e2πi0.x1x0 |1〉)

⊗ (|0〉+ e2πi0.x2x1x0 |1〉)...(|0〉+ e2πi0.xn−1...x0 |1〉)

=

N−1∑y=0

Ay |yn−1...y0〉

(180)

This identity can be understood as a series of phase factors:

Ay = e2πi(yn−10.x0+yn−20.x1x0+...+y00.xn−1...x0). (181)

One can verify using this identity that the QFT does what we want it to do.

Remarks

1. The QFT takes a product state into another product state. It is not an entangling operation. However,there is some subtlety: the phase of one qubit depends on the input values of other qubits.

2. The operation we have created is a generalization of Hn acting on the block of n qubits. Indeed, theHadamard operation acting on the kth qubit belongs to this family:

H |xk〉 =1√2

(|0〉k + e2πi0.xk |1〉k) (182)

When H acts on |0〉k, it produces |+〉, but when it acts on |1〉k, it produces |−〉.

3. The phase operator Rd will imprint a phase on a qubit:

Rd =

(1 0

0 e2πi/2n−d

). (183)

43

4. We can put together the QFT from Hadamard and Rd gates. The QFT quantum circuit is:

|xn−1〉 H Rn−2 Rn−3 . . . R1 R0 . . . . . .

|xn−2〉 • . . . H Rn−2 . . . R2 R1 . . .

|xn−3〉 • . . . • . . . H . . . R3 R2

.... . .

|x1〉 . . . • . . . • . . . • H Rn−2

|x0〉 . . . • . . . • . . . • • H

The phase on qubit 4, Φ4, has a contribution of 2π x3

2 from qubit 3, a contribution of 2π x2

22 from the secondqubit, and so on. Note that the circuit is not complete. In order to ensure consistency in the orderingof the bits (most significant bit at the top, down to least significant bit at the bottom), we need to swapdifferent qubits between each other, using the operation SWAP |x〉 |y〉 → |y〉 |x〉. This is typically denotedin circuit form as

|x〉 × |y〉

|y〉 × |x〉This can be constructed as a sequence of three CNOTs, as shown here,

|x〉 • • |y〉

|y〉 • |x〉

which can be proven rigorously as an exercise. Note that we can swap the qubits either before or afterthe conditional rotations - it does not matter, it just reorders the qubits to ensure consistency betweenmost-significant and least-significant bits.

5. We can count the total number of steps, and see that we need at most

n(n+ 1)

2+n

2∼ n2 (184)

operations, where the first term corresponds to the conditional rotations, and the second term accountsfor the swap operations. We see that this is an exponential speedup compared to the FFT, which isO(n2n) (remember that n = log2N).

6. If we look at these conditional rotations Rd in detail, in particular for small d, we see that these rotationsimplement a very small phase ∼ 2−n. The circuit we have drawn is an exact quantum Fourier transform,but we can make an approximation by neglecting gates with distance > m, which introduces an error atmost n2−m.

7. One can also simplify the circuit by noting that controlled-phase gates are symmetric with respect to ex-changing the control and target, since the gate implements |11〉 → |11〉 eiφ. In other words, it does notmatter which qubit is the control, and which qubit is the target, regardless of the phase accumulated. Wecan then replace the role of the control and target qubits in our original circuit:

|xn−1〉 H • • . . . • • . . . . . .

|xn−2〉 Rn−2 . . . H • . . . • • . . .

|xn−3〉 Rn−3 . . . Rn−2 . . . H . . . • •...

. . .

|x1〉 . . . R1 . . . R2 . . . R3 H •

|x0〉 . . . R0 . . . R1 . . . R2 Rn−2 H

Suppose we simply measure the qubits after the QFT. Now, if we start by measuring the first qubit

44

|y〉0 → y0, we know exactly how to apply the phases to the next qubits, without having done any two-qubitoperations. In particular, instead of applying the conditional Rn−2 rotation to the next qubit, assume wemeasure the top qubit, giving us the value y0. Now we can just apply the single qubit rotation

|x〉n−2 → H(Rn−2)y0 |x〉n−2 = |y〉1 . (185)

Next, we can use this result to show that the next qubit will rotate in the following way

|x〉1 → H(Rn−2)y1(Rn−3)y0 = |y2〉 . (186)

We can continue this up the chain, and what we find is that we do NOT need any two-qubit operations!This is in line with our intuition that the QFT operation does not actually entangle the qubits. Remember,it takes a product state and transforms it into another product state, so it is not surprising that this ispossible. All that is required is single qubit manipulations, as well as the feed-forward procedure requiredto implement gates of the form of (185) and (186).

3.5 Quantum phase estimation

Quantum phase estimation is an application of the quantum Fourier transform, that was developed by Kitaev.It is used for, e.g., finding eigenvalues of a Hamiltonian. It was developed after Shor’s algorithm, but it is usefulto understand prior to Shor’s algorithm, since Shor’s algorithm, at it’s core, relies on it.

Consider a unitary U with an eigenstate |u〉. For example, this unitary could result from Hamiltonianevolution,

U = eiHt/h, (187)

where the eigenstates of the Hamiltonian H will be eigenstates of U , and the eigenvalues of the Hamiltonian(energies) will be proportional to the acquired phases

U |u〉 = e2πiφ |u〉 . (188)

Remember: if we write down the energy spectrum of the Hamiltonian H |u〉 = E |u〉 we accrue a phase of2πφ = Et

h under free evolution. In general, the eigenvalues of U can always be written as eiφ, since we knowthat U†U = 1, such that the eigenvalues of U† will be e−iφ. The goal of quantum phase estimation is to find thephase φ, which can be approximated using k bits in binary notation 0.φk−1φk−2...φk, given a unitary matrix Uand eigenvector |u〉. The quantum phase estimation algorithm is given by the circuit,

|0〉k−1 H . . . •

QFT−1

φk−1

... . .. ...

|0〉2 H • . . . φ2

|0〉1 H • . . . φ1

|0〉0 H • . . . φ0

|u〉 / U U2 U4 . . . U2k−1

where the slashed wire next to state |u〉 indicates that it is possibly a multi-qubit register (and thus, U canbe a multi-qubit gate).

We prepare our k input qubits in the superposition state H⊗k |0〉. These k qubits act as control qubitsimplementing these controlled unitaries on our data register, which is in the subspace of U . Next, we will use the

kth qubit to implement conditional U2k−1

. Consider the case where the data register is initialized in |u〉. Theunitary U will result in accumulation of the phase e2πiφ. Thus, we can describe the effect solely on the state ofthe control qubit, which transforms

|0〉+ |1〉√2

→ |0〉+ e2πiφ |1〉√2

. (189)

45

Similarly, the second qubit will transform as

|0〉+ |1〉√2

→ |0〉+ e2πiφ(2) |1〉√2

, (190)

continuing to the kth qubit which picks up the phase

|0〉+ |1〉√2

→ |0〉+ e2πiφ(2k−1) |1〉√2

. (191)

Clearly, the data register is completely unchanged since it is an eigenstate of U2j . Going back to our Hamiltonianevolution, we can see that this evolution is obtained by evolving for times t, 2t, 4t, ..., 2k−1t conditionally on thestate of the relevant data register qubit.

At this point, we need to implement the inverse quantum Fourier transform. To see why this is useful, wecan explicitly consider the output of these conditional unitaries:

|ψI〉 =1

2k/2( |0〉+ ei2πφ2k−1

|1〉 )⊗ ...⊗ ( |0〉+ ei2πφ20

|1〉 )⊗ |u〉 . (192)

Now, we can rewrite this expression using binary fractions. Consider, for simplicity, the special case whereφ = 0.φk−1...φ0 exactly. In other words, φ < 1 can be expressed with exactly k digits. Looking at the first phasefactor in (192), we see that

2π2k−1φ = 2π(φk−12k−2 + ...+φ0

2) = 2π

φ0

2+ 2πm = 2π(0.φ0) + 2πm, m ∈ Z. (193)

We see that only the φ0 term contributes, since the rest are integers and are multiplied by 2π, and can beneglected. We can continue this analysis for all terms in (192). For example, considering the next term, we willhave contributions from the last two digits

2π2k−2φ = 2π(0.φ1φ0) + 2πm, m ∈ Z. (194)

Now we can rewrite (192) in a form that is extremely familiar from our definition of the quantum fouriertransform:

|ψI〉 =1

2k/2( |0〉+ ei2π(.φ0) |1〉 )⊗ ...⊗ ( |0〉+ ei2π(.φk−1...φ0) |1〉 ), (195)

where we have now dropped the irrelevant register |u〉. We can see that this is exactly the quantum fouriertransform of the register |φk−1...φ0〉, in other words,

QFT−1 |ψI〉 = |φk−1...φ0〉 . (196)

Now, all we have to do is measure the k qubits in the computational basis, and we get all digits of φ at once.

Remarks

1. Consider the case of a general phase φ which cannot be exactly represented in terms of k digits. Quantumphase estimation gives a very good approximation to φ, with very high probability. We can rewrite theintermediate state prior to the inverse QFT

|ψI〉 =1

2k/2

2k−1∑x=0

ei2πφx |x〉 , (197)

where we have now transformed to decimal integer value x ∈ [0, 2k−1]. Now we can perform the inversequantum fourier transform and get

QFT−1 |ψI〉 =1

2k

∑y

∑x

ei2π(φx− xy2k

) |y〉 . (198)

46

We can now consider the probability of getting a certain outcome y,

P (y) = |ay|2=1

22k

∣∣∣∣ 2k−1∑x=0

ei2π(φ− y

2k)x

∣∣∣∣2. (199)

This is a geometric series which can be evaluated to

P (y) =1

22k

∣∣∣∣r2k − 1

r − 1

∣∣∣∣2, (200)

where r isr = ei2π(φ− y

2k). (201)

Now we can write y and φ in binary notation as

y

2k=y0

2k+ ...+

yk−1

2

φ =φk−1

2+ ...+

φ0

2k+ δ,

(202)

where δ is the truncation error from expressing φ with only k bits. Now, looking at (200), we see thatthis is a sharply peaked function φi = yi and δ → 0. Formally, one can show that P (y) ∼ 1 requires thatδ 1

2k. In other words, on average we will, with probability exponentially approaching 1, get the outcome

that is exponentially close to the correct phase.

2. Consider the example of k = 3, shown in the following circuit.

|0〉2 H • H • •

|0〉1 H • R1 H •

|0〉0 H • R0 R1 H

|u〉 / U U2 U4

Looking at the top qubit, we see that we prepare a superposition state, and apply the controlled phaserotation

U4 |u〉 = ei2π(4)(φ22 +

φ14 +

φ08 ) = ei2π

φ02 . (203)

(Note that the overall operation looks a bit like an interferometer, where we prepare a superposition, applya phase, and then return back to the original basis.) Now, if we measure this phase, we see that we willmeasure |0〉 if φ0 = 0, and |1〉 if φ0 = 1. So this circuit simply measures the least significant digit of φ.

This is a bit surprising - we would expect to begin by measuring the most significant digit. But this isthe clever aspect of quantum phase estimation! Now, we can move onto the next qubit, which will now besensitive to both the least significant digit, and the second least significant digit:

U2 = ei2π2(φ22 +

φ14 +

φ08 ) = ei2π(

φ12 +

φ04 ). (204)

However, the controlled rotation from the first qubit, as part of the inverse QFT, exactly compensates forthe value of the least significant digit, allowing the second qubit to measure the second-least-significantdigit precisely. Continuing along this line of reasoning, we can see how the QFT is used to measure eachsubsequent digit of the phase very efficiently in this unintuitive order.

3. This algorithm is very useful for estimating energies of eigenstates of a Hamiltonian.

4. However, in general, controlling the Hamiltonian evolution required for this algorithm is challenging toexperimentally implement.

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5. Because of the sequential nature of the measurement, often times we don’t need many qubits to estimatea phase with high precision. It turns out, with only k = 1, we can already measure many digits withhigh precision. The idea here is that we can reuse this qubit, since we are able to perform operations in asequential manner. We first use the qubit to measure φ0 by using a long time evolution. Then we measureφ1 by cutting the measurement time in half, and so on.

3.6 Order finding and factoring

The order-finding and subsequent factoring algorithm is a canonical (and rare) example of an application ofquantum computers. Certain cryptography protocols (e.g., public key cryptography) leverage factoring, a com-putationally challenging problem, to generate a secure key. We will show below that quantum computers can beused to factor large numbers more efficiently than classical computers, and quantify the speedup. Order findingand factoring is also another application of the QFT. In this section, we will describe the order finding andfactoring algorithms in detail.

Consider two positive integers a and N such that a < N , and which have no common factors. We call r anorder of amodN if r is the smallest integer such that ar = 1modN . This is equivalent to

ar = bN + 1, (205)

for b ∈ Z. As a simple example, consider a = 5, N = 44. We can brute force check r = 2, 3, .. to check which r isthe order. In this case, we find that r = 5 is the order, since 55 = 3125, and we see that 3125 − (71 × 44) = 1,satisfying (205). For large numbers, this procedure is computationally challenging. We can get some intuition forwhy in our original approach for finding the order r, the simple example above, where we “brute-force” checkedthe values of r in ascending order.

Mathematical remarksOrder finding is very closely related to factoring. In the factoring problem, we are given the large number N ,

and we need to find its factors p, q where N = pq. Of course, if we know p or q, finding the other factor is easy.However, if neither is known, this is believed to be hard classically. This problem serves as the basis for publickey cryptography (such as RSA, which is widely used today). The basic idea of RSA is that Alice can choosetwo values p and q, send N to Bob, and to decrypt the message, one must know either p or q. A few statementsare in order:

• The fastest known classical algorithm for factoring scales with O(N1/3), where N is exponential in thenumber of digits, or number of bits required to encode the number.

• It is actually believed that faster classical algorithms exist, but they are not yet known.

• In fact, the problem itself is not computationally hard on a quantum computer, since we will see that Shor’salgorithm, reducing factoring to order finding, can solve the problem in a number of steps that scales withlogN .

In fact, in the following we will describe an algorithm which shows that order finding is equivalent to factoring.We consider a large number N to be factored. The steps are as follows:

1. Pick some a < N , and check if the greatest common denominator is 1. This can be done using the Euclidalgorithm, where if we assume N = pq, and a = pz, and that an integer b1 exists such that:

N = b1a+ r1, (206)

then r1 is also divisible by p.

2. We can continue the Euclidean algorithm another step:

a = b2r1 + r2, (207)

where we find that by the same logic r2 is also divisible by p if N and a are divisible by p. Recall that forfactoring, we need to find these two numbers which have a common factor different than 1. Therefore, ifafter many steps we find p 6= 1, our problem is solved.

48

3. However, if p = 1, we must find r such that (an order-finding problem) ar is

ar = bN + 1. (208)

If r is odd, we have to repeat this procedure from the beginning (picking another a).

4. If r is even, we can write this expression as

(ar/2 + 1)(ar/2 − 1)

= bN

= bpq.

(209)

At this point, one should check if (ar/2±1) is divisible by N . If it is, then we must start from the beginning.

5. However, if not, (ar/2 ± 1) shares a common divisor with N . At this point, one can use the efficientEuclidean algorithm to find the divisor.

In other words, if we can efficiently find r that satisfies (205), we can use the Euclidean algorithm to find thefactors of N efficiently. So the problem of factoring indeed reduces to order finding.

Now, with this mathematical remark aside, let us consider the quantum algorithm for order finding. Our approachwill be based on quantum phase estimation. We begin by constructing a unitary

Ua |y〉 = |aymodN〉 , 0 ≤ y ≤ N − 1. (210)

Consider the eigenstates of Ua, which can be formally written in the following form

|us〉 =1√r

r−1∑m=0

exp

(− 2πism

2

)× |ammodN〉 , 0 ≤ s ≤ N − 1 (211)

where r is the order as seen above. We can show formally that this is true by acting Ua |us〉, which will simplylead to

Ua |us〉 =1√r

r−1∑m=0

exp

(− 2πism

2

)× |am+1modN〉 , (212)

where we can now rewrite the terms in the sum using the fact that r obeys ar = 1 from (205), so that the lastterm |armodN〉 can be rewritten as |1〉. We can rewrite (212) as

Ua |us〉 = exp

(2πis

2

)|us〉 , (213)

so we see that |us〉 are all eigenstates of UA with eigenvalue given by exp( 2πis2 ). At this point, we can implement

order finding using the following steps:

1. Prepare some eigenstate (or superposition of eigenstates)

2. Perform QPE to find s/r. With k auxiliary qubits, we can determine with high accuracy k digits of thefraction s/r.

3. Extract s and r, which are integers, from the ratio. This is a bit subtle, but suppose we know the fractionalnumber s/r exactly (e.g. 0.153 in decimal). We can easily write this as 153

1000 , and simplify to get s and r.The subtlety arises from the fact that we have an approximation to s/r, but it turns out since we knowthis with exponential accuracy, we can use so-called continued fractions to obtain very good estimates of sand r.

4. Once we have a vaule of r, we can easily check that it is an order.

Remarks

49

1. How do we prepare |us〉? Let us consider the superposition

1√r

r−1∑s=0

|us〉 =1

r

r−1∑m=0

r−1∑s=0

exp

(− 2πism

r

)|ammodN〉 . (214)

If s 6= 0, each term will have a phase, these terms will interfere destructively and add to 0. So we canrewrite (214) as

r−1∑m=0

δm,0 |am,modN〉 = |1〉 . (215)

This means that we can actually produce the superposition in (214) by simply preparing the binary state|1〉, and run this phase estimation. Then, for different instances of phase estimation, we will sample differenteigenstates for UA.

To summarize this approach, we will start in |1〉 and run phase estimation several times. In the firstinstance, we will estimate s1

r , then s2r , etc. Knowing a few si/r will help us determine the simplified

expression sr .

2. It remains to show that we can implement QPE efficiently. This is done using modular exponentiation.Recall that we need to implement e.g.

U2a |y〉 = |a2ymodN〉

U4a |y〉 = |a4ymodN〉

. . .

(216)

Consider the register |x〉 |y〉, where |x〉 is the auxiliary register we use to implement QPE, and |y〉 storesthe superposition of eigenvectors to Ua. We want to implement the transformation

|x〉 |y〉 → |x〉Uxk2k−1

a ...Ux020

a |y〉

= |x〉 |axk2k−1+x020

ymodN〉= |x〉 |axymodN〉.

(217)

If we can implement |1〉 → |amodN〉, then |1〉 → |a2modN〉, etc, then the entire procedure will only requirek steps to implement the unitary to the power 2k. What remains is to show how to implement this modulosquare. This is similar to conventional multiplication and requires only n2 operations. This step is a bitcumbersome, but is related to finding a classical multiplication algorithm/circuit and making it reversible.At this point, one can implement it using e.g. Toffoli gates. In total, this requires kn2 ∼ n3 ∼ (logN)3.

3. We can consider starting with

H⊗k |0...0〉 |1〉 =∑|x〉 |1〉 , (218)

and after the modulo square and subsequent exponentials, we get∑|x〉 |axmodN〉 . (219)

Note that r is exactly the period of the function (ax mod N). At this point we can perform QFT. We cansee the effect of this by writing the expression in (219)∑

m

(∑j

|m+ jr〉)|ammodN〉 . (220)

Now we see that QFT will exactly reveal this period r. (More precisely, QFT will give us a very preciseestimate of r.)

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3.7 Implementing quantum algorithms

Errors are kind of like a virus - they spread exponentially. We wonder if there is any hope to build a quantumcomputer with many qubits that is of practical use, in the presence of errors. There are some ideas developedtheoretically that allow for this. The cycle is development of algorithms, to building, to fighting noise, and thendesigning algorithms that can work in the presence of noise - co-design is the idea of thinking of this problemwith all of these steps together. If we start coming up with new algorithms, what kind of means do you needin a lab to implement them? An important concept in this field is universality. It turns out that any quantumcomputation can be built from a discrete set of operations. The simplest and best-known approach is based onquantum circuits, which we have already considered. Today, we will show that any quantum computation canbe built from a finite set that is a so-called universal set of quantum gates.

3.7.1 Universality theorem

To begin, we will prove the theorem. The theorem states that any U on a dxd dimensional system can bedecomposed into U = U1U2...Uk with Ui acting only on two states, and where k ≤ d(d − 1)/2. The idea of the

proof (see Nielsen & Chuang) is to construct U†kU†k−1...U

†1U = 1, and find the Ui. Suppose we have a 3x3 matrix:

U =

a d gb e hc f j

(221)

We find U†1U = 1, such that b→ b′ = 0, such that:

U†1 =1√

|a|2+|b|2

−a∗ b∗ 0b −a 00 0 1

(222)

Next we find U†2 such that U†2U†1U = 1

U†2 =

1 0 00 ... ...0 ... ...

(223)

For a d-dimensional matrix, we use the same procedure and find that order d2 steps are needed. However, thereis a problem: d scales exponentially with the number of qubits, such that the total number of operations wouldscale exponentially with the number of qubits. Next we will like to construct two-level matrix corresponding toarbitrary states from single and two qubit gates. Suppose we want to construct a unitary between two states|101001〉 and |110011〉. Note that the operations should not influence any other states, such that this is onlyinfluences the two states chosen. Our method is the following: we would like to take the first state and flip bitsone by one in such a way that only one bit changes at a time, conditional on the other bits. At each step there isa controlled flip so that only each bit is flipped if the state corresponds to the chosen starting and ending states.That way we can stay within the subspace of the two states chosen. We can therefore get arbitrary rotationamong these two many-body states using these controlled rotations. This requires only linear in the number ofqubits n, so we can implement an arbitrary 2-level unitary using only ∼ n steps.

For example, recall that the gate CnZ = 2(n− 2)C2Z which is linear in n. Thus each two level U is at mostn2 single and two qubit gates.

3.7.2 Discrete set of universal operations

The following set of gates is universal: H, CNOT, and T, where we have the T gate:

T =

(1 00 eiπ/8

)(224)

The rotation by π/8 is important to get a rotation away from one of the principal axis. It turns out that thisis sufficient to approximate any single-qubit rotation, as will be proved on the homework. The angle of rotationcos θ/2 = cos2 π/8 can be generated, which is an irrational fraction of π, and that can be used to generateany rotation on the Bloch sphere. The Solovay-Kitaev theorem shows that the number of rotations is actually

51

logarithmic in 1/ε, where ε is the error. However, for an arbitrary unitary, this will require an exponential numberof gates. The art of quantum algorithms is to find algorithms such that they scale polynomially in the numberof qubits n. This is the basis of quantum complexity classes.

3.7.3 Other approaches to quantum computation

The gate model is universal, but it is by no means unique: there are other approaches to implementing quantumalgorithms and building quantum computers. One approach is to build a machine with a complicated rangeof interactions that implements the Hamiltonian that corresponds to the desired unitary U . If we succeed inbuidling this Hamiltonian, it is relatively easy to implement the desired evolution, but generally it is not easy tobuild the desired Hamiltonian. In general, the Hamiltonian will have products of k qubits, with k running from1 to n. Such terms are called ‘k-local’ terms, and the Hamiltonian becomes very complicated. If we can do this,what will be the power of this machine? Can it be universal? We will discuss this point.

Quantum computing by measurement is another approach. If you start with a given state, such as a 2D clusterstate, and perform sequential measurements on it. For example, by performing measurements and single qubitoperations on columns of a 2D cluster state, effective operations are performed on the remaining qubits. If themeasurements are started to the left, the physics of the effective operations on the remaining bits is like time inthe circuit-based model propagating to the right. This approach is also universal.

Examples

1. Quantum simulations. Suppose you are given a Hamiltonian and you would like to make some predictions:what is the ground state? What is the energy of the ground state? What are the quantum dynamics afterturning on the Hamiltonian and starting in a particular state (called a quantum quench)? These problemsare hard classically, because it requires solving 2n differential equations, corresponding to the dimensionof the Hilbert space. This is quite hard for e.g. n > 50. Note that if there are no interactions (or moreprecisely, specific types of interactions), the number of equations can be linear in n and this can becometractable, but in general, it is not. There are two ways to answer these questions. One is to prepare theHamiltonian and the state in the desired system and let it run the dynamics under the Hamiltonian, thenperform measurements. However, if you want to measure a more subtle quantity like the energy of theground state, you can combine this approach with quantum phase estimation. Current quantum computerswhich have a number of qubits on the order of 50 are already useful for this application. This simulationidea is how the field of quantum computers started - Richard Feynman stated that we better use quantumsystems to simulate quantum systems!

2. Adiabatic algorithms can be best formulated in terms of solving combinatorial optimization problems suchas NP complete problems like the Traveling Salesman problem, MaxCut, and Maximum Independent Set(MIS). These problems attempt to find a minimum of a cost function of a certain number of variables.Suppose these variables are bits (classical problem). You can write this cost function in terms of the qubitstates and solve them using a quantum computer. Often, terms in the cost function depend on products ofmultiple states, which correspond to a many-qubit interaction. Some of these problems can be describedusing a graph - suppose that we have a number of vortices and they are connected by links. The maximumindependent set is the maximum number of vertices that are not directly connected by the links. Theproblem of MIS involves finding the maximal number of these vertices, without violating this independentset constraint. This problem is very easy to formulate and check for a given possible solution, but itis an NP complete problem (for some examples it is even harder than NP complete). Even finding anapproximate solution is quite hard. One approach where quantum computers can help is to design a costfunction H(z1, ...zN ) as a Hamiltonian corresponding to the cost function which has a form depending onthe clauses (constraints), for example Hp = aZ1Z2 + bZ2Z3Z4 + cZ4Z5 + ... with penalization a, b, c, ...corresponding to the clauses. The ground state corresponds to the lowest value of h, the cost function- which is solving the goal of the problem: Hp |Z1...ZN 〉 = h(Z1, ...ZN ) |Z1...ZN 〉, simulating the groundstate and measuring it guarantees that we solve this problem. This makes a correspondence between hardproblems in computer science to finding ground states of complex spin models and statistical mechanics.Now, with this corresponding established, we can try to design a new family of algorithms, called adiabatic

52

quantum algorithms. Consider a situation where the Hamiltonian depends on time, such that at t = 0, theHamiltonian has a ground state that is easy to prepare, for example a

∑i xi. Once we prepare the state

|−〉, we know we are in the ground state. By changing the Hamiltonian to the desired Hamiltonian Hp

over a long time T , we are guaranteed to be in the ground state, as long as the change is performed slowlyenough. How slow is slow enough? The time T must take longer than the energy gap, such that the Fourierlimit of this time is smaller than the energy gap. However, the gap in the spectrum generically closesexponentially for complicated Hamiltonians including any arbitrary interactions, such as a spin glass. Theadiabatic quantum computer can be shown to be universal, meaning it is equivalent to the circuit modelwith polynomial overhead.

3. Variational quantum algorithms. Two examples we will consider are called Quantum approximate opti-mization algorithm (QAOA) and Variational Quantum Eigensolver (VQE). QAOA is targeted for classicaloptimization problems, whereas VQE is designed for solving hard quantum mechanical problems.

The idea behind variational quantum algorithms is a generalization of the adiabatic algorithm. Recall thatthe adiabatic Hamiltonian is

γHp + β∑i

Xi. (225)

Similarly, we begin by defining unitaries

Up(Hp, γ) = e−iγHp

Ux(β) =

n∏i=1

e−iβiXi(226)

and preparng qubits in the symmetric superposition

|s〉 =1√2n

∑z

|z〉 , (227)

which is obtained by performing H⊗n |0〉⊗n. This implies that Xi |s〉 = |s〉, in other words |s〉 is aneigenstate of each individual Pauli X operator with eigenvalue 1.

In QAOA, we apply UpUx k times, giving us

|~γ, ~β〉 = Uxk(βk)Upk(γk)...Ux1(β1)Up1(γ1) |s〉 . (228)

Here, we measure the cost function and find new ~γ, ~β, and optimize over these vectors (trajectories) inorder to minimize the cost function. Here, k is an effective circuit depth, and as k → ∞, this convergesto the adiabatic path. However, this is a more general approach since we can in principle reach the costfunction minimum faster (in a fininte number of states).

Remarks

(a) QAO (adiabatic limit), QAOA are examples of heuristic algorithms. This means that we do not yethave any guarantee that the algorithm will actually provide a speedup. The general belief in thefield is that we must now build quantum computers to test how they will perform. This may soundincredible, but this was certainly the case for numerous classical algorithms on ordinary computers.One prominent example of this are so-called Deep Neural Networks, which solve hard problems quickly,yet noone knows why they work so well.

(b) QAOA has close connections with the gate model of quantum computing, since Up(γ) can be builtfrom 2-qubit gates.

The current frontier of QAOA is interested in tackling the following problems

• How expensive is it to encode a given, useful Hamiltonian? What is the overhead in implementing theoptimization?

53

• Can we make use of hardware implemented n-qubit gates? Does this map effectively to n-local models?

• Does the performance depend on the hardware implementation (e.g. gate connectivity)

• How can we design the algorithm specific to the hardware implementation (co-design)

4 Implementation of quantum computers

In the early 2000’s, David DiVincenzo set forth general criteria for a quantum computer, based on the idea ofuniversality. These are known as DiVincenzo criteria:

• A set of well-defied qubits with long coherence times

• The ability to initialize all qubits in a simple state

• The ability to individually control and measure qubits

• The ability to perform two-qubit operations, including the ability to turn the required interactions on andoff

• Need to be able to scale qubits, such that adding qubits can be done with resources ∼ O(n)

It turns out that almost any non-harmonic quantum system can be used as a quantum computer! In the 2000’s,this resulted in a lot of creative ideas and proposals for building quantum computers (in fact, even some basedon Harmonic systems).

In practice, DiVincenzo’s criteria turned out to be only a guide. In today’s modern research, we often seethe terms NISQ (Noisy intermediate-scale quantum era - the era we currently live in), as opposed to FASQ(Fault-tolerant application quantum era - hopefully a near-future era). The issues which have emerged is thatindividual qubits (or a few qubits) can be controlled and operated with exquisite precision. However, often evenindividual gate fidelities scale with n - in other words, it becomes harder to build a high-fidelity machine whenthere are more qubits to control. Physical systems may employ collective gates on large registers, which can beadvantageous in some situations, but can also be a hindrance in implementation of simple single and two-qubitgates. The ability to perform two-qubit operations between certain qubits (i.e. connectivity) is yet anotherlimitation. Finally, the ability to control and readout all qubits in parallel with high fidelity is a major challenge.A summary (by no means comprehensive, or up-to-date!) is presented in Fig. 9

4.1 Neutral atoms and ions: background

These are individual atoms or ions suspended in vacuum “tubes” or chambers, which have very good quantumcoherence when individually isolated and cooled. These, in fact, form the best timekeepers (atomic clocks) whichare the basis for time standards and global positioning systems (GPS). A major challenge is to isolate and ma-nipulate single atoms in a way that can be scaled up to many atoms. A second major challenge is to engineerthe required controllable interaction to make two-qubit interactions.

Summary of atomic physics

A generic atomic level structure can be understood by simply studying the quantum mechhanical problemof the Hydrogen atom: a single negatively charged electron orbiting around a single positively charged proton.By solving the Schrodinger equation for the potential between these particles, we obtain an energy spectrum ofdiscrete levels, which are conventionally described by several quantum numbers |n, l, s, I,ml,ms,mI〉, where n isthe principle quanutm number (electronic orbital level), l is the angular momentum quantum number, s is theelectronic spin degree of freedom, I is the nuclear spin sublevel, and the mi are the individual values for thevarious angular momenta.

We can see that this results in a fairly complex spectrum! Each principle orbital will be split by the variousinternal spin degrees of freedom into several levels. In the case of Hydrogen, we have a spin-1/2 electron andspin-1/2 nucleus, hybridization of which (via hyperfine interaction) results in the formation of triplet and singletstates (a total of 4 magnetic sublevels). The hyperfine interaction is a type of dipole-dipole interaction, whichprimarily results from the contact term (overlap between the wavefunctions of electron and nucleus). The furthersplitting of levels is a result of the Zeeman interaction with an external magnetic field. Once the spectroscopy

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Figure 9: High-level overview of some leading quantum computing systems, adapted from Monroe et al, Science(December 2016).

of the individual atom is known, we generally choose a qubit consisting of two “spin” states which have the bestcoherence between them (for example, two states in the electronic ground state with different F or mf , whichcan be manipulated using e.g. radio-frequency fields).

The main tool for manipulation of atoms and qubits therein is the laser. These can be used to implementcoherent rotations of qubits, perform cooling and trapping of the atoms themselves, as well as preparation inwell-defined states via optical puumping.

Examples:

1. Optical pumping. The atom starts at room temperature, which means that it is with equal probability inall of the ground states. By using laser light with a circular polarization σ+, which always increases themagnetic quantum number by +1, we can ensure that that on average, the magnetic quantum number ofthe atoms is increasing under the laser field. The atom will continually be excited to a state with +1 angularmomentum, and decay probabilistically to a state with either ±1 or the same angular momentum quantumnumber. After a few excitation and spontaneous emission cycles, the atom is stochastically polarized intothe level which is no longer coupled to the laser field, since all of the sublevels that were coupled to thelaser field were “optically pumped.” Once polarized with high probability, we have effectively cooled theinternal degrees of freedom of the atom to very low temperature (T → 0). Where does the entropy go?

2. The physics of atomic motion in laser light can be understood in a similar spirit. Intuitively, the absorptionand emission of a photon changes the atomic momentum by ±hk. Mathematically, we can describe theemission of a photon using the Pauli lowering operator between the ground |g〉 and excited |e〉, σ− = |g〉 〈e|.The running wave can be descibed in space as Ω(x) = Ω0e

ikx. The Hamiltonian that arises will be givenby

H = hΩ0eikxσ+ + hΩ0e

−ikxσ−, (229)

where the two terms correspond to stimulated absorption or emission of a photon from the classical laser

55

field. By rewriting the exponential in terms of a quantized position operator x

eikx =

∫dpdp′ |p〉 〈p|eikx

′|p′〉 〈p′|

= dp |p+ hk〉 〈p| ,(230)

where we have recognized that 〈p|eikx′ |p′〉 = δ(p− p′+ hk). So we can rewrite the Hamiltonian very simplyas

H = hΩ0

∫dp |p+ hk〉 〈p|σ+ + h.c. (231)

where h.c. denotes the Hermitian conjugate. Hence, an atom can transition between |p〉 |g〉 ↔ |p+ hk〉 |e〉.

Remarks

• When this process is accompanied by spontaneous emission, which results in irreversible momentumtransfer, we can achieve either cooling or heating. Depending on the configuration of the laser fieldand the atoms, this is the basis for so-called Doppler cooling.

• In the off-resonant case, we can think of the interaction in terms of dipole forces. Recall that we willhave an effective Hamiltonian (obtained via perturbation theory in small parameter Ω0/∆, where ∆is our detuning from the optical transition

Heff =|Ω|2

∆|g〉 〈g| (232)

where Ω depends on the position of the atom! This creates a potential and can in principle be usedto trap the atom at the location of the optical field maximum, for example in the central focal pointof a tightly focused “tweezer” beam.

Remark: simple harmonic oscillators (SHO)The simple harmonic oscillator is a good approximaton for the motion of trapped ions and atoms, assuming

the atoms are cold and exploring the bottom of their trapping potential (which is usually quadratic, to leadingorder). Additionally, the SHO captures the quantum description of electromagnetic fields, down to the level ofsingle photons.

Recall the conventional Hamiltonian for the SHO

H =p2

2m+ k

x2

2. (233)

We can quantize this Hamiltonian by introducing quantum mechanical operators p→ p and x→ x which do notcommute, such that [x, p] = ih. More conveniently, we can write superpositions of position and momentum interms of the so-called annihilation and creation operators

a =1√

2mhν(mνx+ ip)

a† =1√

2mhν(mνx− ip)

(234)

where we can clearly write

x =

√h

2mν(a† + a)

p =

√h

2mν

a† − ai

(235)

X IN TERMS OF A, Adag

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Now we can rewrite the Hamiltonian in terms of the creation and annihilation operators, instructively as

H = (a†a+1

2)hν, (236)

which clearly has eigenstates of the form a†a |n〉 = |n〉, where |n〉 are known as Fock states. This Hamiltonianhas a ground state (n = 1) with nonzero energy E1 = hν

2 , and a ladder of harmonically spaced energy levelsseparated by hν. This is clearly different from the case of a qubit, since if we try to drive one transition, we driveall transitions because of their equal energy spacing. So it is clear that we cannot simply treat this system theway we treat a qubit, under any approximation.

The purpose of the creation and annihilation operators are also clear in this picture. If we apply them to theFock states, we see that they “raise” and “lower” the Fock state respectively by exactly 1 quanta:

a |n〉 =√n |n− 1〉

a† |n〉 =√n+ 1 |n+ 1〉

(237)

Note that these prefactors are important since, application of a†a should obey the eigenvalue equation a†a |n〉 =|n〉.

4.2 Trapped Ion Quantum Computer

This is an important sytstem to study, as it was one of the first systems that convinced the field that quantumcomputers could actually be built in the laboratory. Additionally, it remains at the frontier of the field of quantumcomputing even today.

This model of quantum computing relies on charged particles trapped in vacuum. Since the particle is charged,we can use electric fields to confine it. We would like to engineer an electrostatic potential which effectively pushesthe ion from all directions to a single stable point, leading to confinement in three dimensions.

However, basic electromagnetic theory tells us that this is not straightforwardly possible! This is impossiblebecause of Gauss’s law

δE = 0, (238)

where we see that if we make a sphere around our ideal configuration, we have violated (238). The best that wecan hope for in reality is confinement in two dimensions, and anticonfinement in three dimensions (i.e. a saddlepoint in potential space). We know however, that the particle will be unstable with respect to perturbations inthis third direction.

The solution is in fact not to use a static potential, but a dynamic one. By rotating the saddle, we can obtainan effective time averaged potential that confines the ion in all directions as desired. This is known as a Paultrap. In order to undrstand this, consider a classical 1D oscillator described by

x = (k2cosΩt)x = 0

k2 =eV0

md2,

(239)

where e is the ion charge, m is the ion mass, V0 is the RF voltage amplitude, and d is the trap-size. We canrecognize (239) as a Matheiu equation, where x(t) is bounded for k Ω.

The trajectory contains secular oscillations at the trap-frequency ωtrap ≈ k2

Ω , typically in the MHz frequencyrange, as well as micromotion oscillations at fast frequnecies Ω in the 100 MHz range. The ion feels an averagedpotential, and assuming the RF frequency Ω is high enough, this enables confinement of the ion in all threedimensions

Many modern ion traps use the configuration consisting of four rods, with positive voltage applied to twoopposing rods, and negative on the alternate two. Using RF voltages, this leads to confinement in the plane ofthe rods. The two ring electrodes are used for confinement along the direction of the rods. Ths is typically usedto generate a chain of ions along the axis of the rods.

The result of these techniques is the trapping of single ions in a confined potential, with relatively high overalltrapping depth, even larger than room temperature. If we can cool the ion towards the bottom of the potential,to a very good approximation, this leads to simple-harmonic oscillator physics describing its motion:

HT =p2

2M+M

2ν2Tx

2 ≡ hνT (a†a+1

2), (240)

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where the effective trapping frequency νT after all averaging over the RF trap is usually a few MHz (comparedto the RF frequency of the order 100 MHz). Quantized motional excitations in this trap are known as phonons.

Lasers are now used to manipulate the motion of the ion in this trap, in a way that is different from thefree-space case. The intuition for this is that the trap can now absorb the momentum kicks from the light,so momentum does not need to be conserved, leading to inelastic atom-photon collisions. As such, light can bedirectly used to modify n, the number of phonons (motional excitations) in the trap. Mathematicallly, applicationof the laser results in the atom-field interaction Hamiltonian

HA−F = hΩ0σ+eikx + h.c., (241)

where we can rewrite the position operator as

x =

√h

2MνT(a† + a) = a0(a† + a), (242)

where a0 is the ground state position of the simple harmonic oscillator. From (242), it is clear that by applyingthe laser field we can also change the positional state of the ion by either adding or subtracting phonons.

In the special case where the ion is strongly confined, such that a0k 1, or a0 λ (the zero-point motionis much smaller than the wavelength), we can expand the Hamiltonian in leading order

HA−F = hΩ0σ+ + hΩ0a0ki(a+ a†)σ+ + ...+ h.c.

≡ HRabi +H1 +H2.(243)

This is known as the Lamb-Dicke limit. The first term just corresponds to Rabi oscillations (elastic scattering)between the ground and excited state of the atom, without affecting the motional state of the ion.

The other terms in (243) correspond to the case where the atom interacts with the field and also changes itsinternal motional state. Taking the terms shown proportional to σ†: these correspond to excitation of the atom,either with absorbtion of an additional phonon (a†), known as the blue sideband, or removal of a phonon (a),known as the red sideband.

Remark: physics of simple harmonic oscillator coupled to qubit

1. Consider the reduced Hamiltonian from (243) in the form

H1 = hg(aσ† + a†σ). (244)

This Hamiltonian corresponds to quantum excitation exchange between the qubit and the simple harmonicoscillator.

Here, we can consider some detuning ∆E between the oscillator and the qubit, such that ∆E = Ee−Eg−hν,for all n. Manipulating this detuning will allow for direct removal of a phonon by laser driving. Furthermore,we see that the coupling rate will actually grow with n as 〈n− 1|a|n〉 ∼

√n! The simple ground state |g〉 |0〉

is clearly an eigenstate. This will be crucial to the idea of cooling to the ground state. However, due tothe interaction in (244), the higher levels containing product state of qubit and oscillator are no longereigenstates, for strong enough coupling g.

As an aside, the atom-photon excitation exchange (as opposed to vibrational phonons) is known as theJaynes-Cummings model of cavity quantum electrodynamics, and is a crucial component in several primi-tives of quantum information processing.

2. Consider the second termH2 = hg(aσ + a†σ†), (245)

in which excitations are created in pairs. This, is known as parametric amplification and can be used togenerate entanglement between excitations within the Harmonic oscillator (i.e. quantum states of motion,also known as squeezed states.

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With this in mind, one can now consider spontaneous emission which has not been included in the Hamiltoniantreatment above. In general, this has been bad for quantum manipulation, but we have seen that it is crucial forcooling and initial state preparation. If we consider the trap frequencies νT ∼ 100 kHz - 10 MHz, we see that werequire extremely low temperatures, necessitating such cooling schemes.

Considering that the excited atom emits in three-dimensions and a particular excited state |e〉 |n〉 decays withthe following pathways

→ |g〉 |n〉 rateγ

→ |g〉 |n+ 1〉 rateγ(ka0)2

→ |g〉 |n− 1〉 rateγ(ka0)2,

(246)

showing that the inelastic transitions will be suppressed in spontaneous emission in the Lamb-Dicke limit.We can now perform laser-cooling of ions by exciting with a red detuned laser, effectively driving the system

from the state |g〉 |n〉 → |g〉 |n− 1〉, reducing the motional quantum number by 1. Now, when we spontaneouslydecay, we know we will most likely decay to the state |g〉 |n− 1〉, leaving us in a lower occupation state than westarted with! This process will continue down the Harmonic ladder until we are in |g〉 |0〉, at which point ourlaser will be detuned from all transitions.

This is the idea of so-called phonon-sideband cooling. Photon absorbtion leads to transitions of the form

|g〉 |n〉 → |e〉 |n− 1〉 , (247)

whereas spontaneous emission predominantly leads to transitions of the form

|e〉 |n− 1〉 → |g〉 |n− 1〉 . (248)

Remarks

1. Cooling is most efficient when γ νT , which is known as the sideband resolved regime, since the linewidthof the transition given by γ is smaller than the separation between the sideband and the carrier (direct)transition. In the other limit, γ νT , cooling can still be performed, and this is known as Doppler coolingas discussed before.

2. The maximum rate of cooling is proportional to γhνT . This goes back to the question of where the entropygoes in this process. The answer is that spontaneous emission carries the entropy away with the emittedphoton. This is crucial: we need a combination of coherent laser driving and dissipation via spontaneousemission in order to perform cooling.

We can also use light to perform measurement of ion internal states. If we consider two ground states ofthe ion which can be used as a qubit, We typically have a cycling transition associated with one of the states|1〉 , suchthatapplicationofthelaserexcitesbetween|1〉 and |e〉 in a cyclic fashion (the population stays within thismanifold). By applying this laser and detecting some fraction of the emitted photons, we can determine thatwe were in the state ket1, since |0〉 would not have resulted in a scattered photon. Thus, in ideal conditions,detection of a photon corresponds to a projective measurement of the qubit in the state |1〉. Realistically, we canplot histograms of the number of photons detected on average from the two qubit states, and as long as they arewell resolved, we can perform effective projective measurements.

Next, we can perform qubit operations on the long-lived hyperfine ground states (often known as “spinstates”). This can be done with radio-frequency illumination to drive Rabi oscillations. However, these rotationsare often done using so-called Raman transitions, which are two-photon transitions using an intermediate excitedstate. By driving the two transitions simultaneously with Rabi frequencies Ω1,Ω2, with a large detuning fromthe excited state ∆, we can obtain effective single-qubit rotations at a Rabi frequency ΩR ∼ Ω1Ω2

∆ . It turns outthe coherence properties of ions prepared in superpositions in such a fashion are very good, now on the order of10 seconds.

Next, in order to perform two-qubit operations, we can rely on the Coulomb interaction between the chargedions trapped in a chain. This is done by coupling the spin state to the common motional mode of ion vibrations.

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By trapping many ions, we see they will arrange with a particular spacing due to the repulsive potential betweenions. Now, ions will vibrate about their equilbrium positions in a way that is coupled to one another (since eachion feels the presence of the other ions via the Coulomb interaction). This can be treated as a system of coupledoscillators - we can solve this similarly to the classical case by finding the normal modes.

We consider the Hamiltonian consisting of N coupled Harmonic oscillators

H =1

2

N∑i=1

Mν2i + ν2

TMx2i +

∑i,j

e2

xi − xj. (249)

This describes N coupled simple harmonic oscillators near equilibrium, and has N normal modes. The lowestenergy mode consists of center of mass motion of all ions

xCM =

N∑i=1

xiN, (250)

with frequency ν = νT and effective mass M ×N . The next lowest mode will have a different ferquency√

3νT ,and so on.

Thus, because each mode has a different frequency, motional modes can selectively be excited with a laserindependently. For example, with N = 2, we can show that we have modes

xCM =x1 + x2

2(251)

for the center of mass, andxrel = x1 − x2, (252)

corresponding to relative motion. This can be captured in a diagonalized Hamiltonian

H =1

2(MCMν

2CM + ν2

TMCMx2CM ) +

1

2(Mν2

rel + ν2TMxrel

2), (253)

where M can be shown to be√

3M .One challenge is that for larger and larger ion chains, the overall trapping frequency needs to be reduced in

order to keep the system stable, in other words, to be able to resolve the modes from one another, which becomesmore and more challenging and complex as we add ions and create additional motional modes.

4.2.1 Approach to two-qubit operations

We have seen before that the ion-phonon interaction is:

Haf = hΩeikxiσ+i + h.c. (254)

We can write the position as:xi = xCM + rest (255)

If the laser is tuned into resonance with the COM motion, the rest can be ignored. The leading order termis then:

Hiaf = HRabi + Ω

ka0√N

(aCM + a†CM )σ+i + h.c. (256)

The Mossbauer effect which yields the 1/√N dependence corresponds to the fact that if we have a big object

with many degrees of freedom and with mass, the entire object absorbs the momentum. Since the photon couplesto the momentum of the entire chain, we have a collective effect which can be used to generate entanglementbetween the ions.

For example, if the laser is tuned to |n〉 |g〉 → |n− 1〉 |e〉, the effective dyanmics is:

HJC = hΩka0√Nσ+i aCM + h.c. (257)

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This is equivalent to a TLS interacting with a cavity photon field, and allows for the quantum state of the ionsto be exchanged with intertial (qubit) states and the motional (phonon) state of the ions! In practice this can bedone with two-photon stimulated Raman transitions (add pic from lecture notes). If the two photon transitionis detuned by the trapping frequency, then the ion will exchange internal and motional states. In particular, fora single ion, we have:

(α |0〉+ β |1〉)⊗ |0〉m → |0〉 ⊗ (α |0〉m + β |1〉m) (258)

If there are more than one ions, then this motional degree of freedom includes all of the ions. The idea is toexchange motional excitations between ions, which come from their internal degrees of freedom, using the ionicmotion as a data bus. For further reading see Cirac and Zoller, PRL 74, 4091, 1995.

State of the art and challenges

• The ion control has been perfected to a high degree. The gate fidelities are better than 99.9%, way aheadof every other system.

• However, scaling the control to a larger system is challenging. The ions heat up due to the coupling to theenvironment, and at large N thre is a large number of modes which can be difficult and slow and isolate asingle one.

• Modern approaches include off-resonant motional excitation, which moves the ion conditionally on whichstate it is in gently. For example, if it only pushes the state |1〉, then it creates a dipole between two ions.This approach is robust since you don’t need the ion to be perfectly in the ground state.

• Geometric gates are also used (see Molmer-Sorensen gate) which are also relatively insensitive to themotional states.

• 1D traps have achieved up to 50 qubits, but 2D traps is challenging. There has been some effort for adecade or so to extend to 2D, but this is also challenging - there are more degrees of motional freedom.

• Another approach is to cool the traps cryogenically to reach the ground state, but that can also be technicallychallenging.

• Some approaches for scaling up involve a ‘quantum CCD’ which includes storage zones and interactionzones, and shuttling the ions around between these zones. The challenges of this is that moving the ionstends to heat them, especially around the corners.

• Another approach involves entanglement by measurement - using photons and measurement to carve outentangled states of multiple ions, and using other ions as registers.

For further reading, see D. Liebfried et. al., RMP 75 281 (2003); R. Blatt & D. Wineland, Nature 453 2008;I. Cirac & P. Zoller, Physics Today, March 2008.

4.3 Neutral atom quantum computer

Neutral atoms are like ions except they do not have a net charge. Why would we want to use them? They areextremely coherent in terms of internal degrees of freedom (information can be stored for a long amount of time- the best atomic clocks use neutral atoms), and there are lots of neutral atoms around, so it is relatively easy tocreate identical arrays of ions. The key challenges is that it is hard to isolate and control individually, and theinteractions are weak (which are needed for multi-qubit gates). The approach is as follows:

• Optical tweezers are traps that use focused laser beams or standing waves. There is a point where theelectric field is maximized at the center. The light, if detuned properly, will induce an electric dipolemoment which pushes the atom to the point of maximum intensity. The dipole potential will be |Ω|2/∆.In this way, an atom can be trapped and held for a reasonably long time in vacuum.

• Additionally, we can start with an array of tweezer traps, and load them. However, the trap frequencies areonly 10-100 MHz, which means that the atoms must be cooled before loading (unlike the case with ions).

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• For interactions, we have two possibilities: (1) bring the atoms on top of each other to create ‘cold collisions’,but this is a slow, delicate process; and (2) excite atoms into ‘Rydberg states’. These are highly excitedstates with principle number n >> 1. When this number is high, the size of the atom becomes large,on the order of microns. This induces dipole-dipole interactions which scale as n4, but the Van de Waalsinteraction (a result of virtual excitations in a dipole-dipole interaction) scales as n11, which is quite a largenumber. These Rydberg states have quite a long lifetime, since they cannot undergo spontaneous emissioneasily, and can last for over 100µs. This combination allows one to do very fast, high fidelity quantumoperations.

• However, neutral atom trapping is a bit more delicate than trapped ions - the traps are relatively shallow,and the limitation is given by the vacuum (collisions with background atoms). Depending on how well thevacuum is created, the atom lasts in the trap for 10s - 10s of minutes. The solution is to reload the atomsoften.

• In general, different atomic states will have different trapping potentials, since the interaction of the laserwith the atom depends on its state. For example, the excited state in an optical tweezer is an anti-trappedstate. This kind of trapping would generally entangle the motional and atomic degreees of freedom, whichis a source of decoherence. The solutions for this involve ‘magic wavelength trapping’, frequencies suchthat the potential is the same for all states, as well as making the gates very fast, so that the atoms do nothave time to move - the time spent in the excited state is very small.

In principle, we can directly use the Rydberg interaction to implement e.g. a control-phase gate. However,there is a more refined technique for implementing gates based on Rydberg interactions known as RydbergBlockade. If atoms are far apart, and we excite the Rydberg transition resonantly, they will simply undergo Rabi

oscillations individually. However, as they come closer together, at some point, the energy shift UV DW ∼ n11r6 will

become very large, and the transition will not be in resonance anymore for both atoms. We can still excite oneof the atoms, but not both because of the significant additional energy cost of the Rydberg interaction, whichleads to a shifted energy of the doubly excited level.

Thus, the simultaneous excitation of two atoms will be blockaded at a particular distance on the order of theblockade radius Rb few µm, with variations between 1 and 100µm by careful choice of Rydberg state and theRabi frequency of Rydberg excitation lasers. One important aspect is that once the atoms are closer than theblockade radius, their exact separation is unimportant, since the blockade will be in effect. This means that highfidelity operations can be implemented in a way that is relatively insensitive to exact atomic position and theirmotional state

The atoms in the Rydberg states are highly excited, meaning they can undergo decay process. However, thetimescale for this decay is usually quite long (several hundred µs), and operations can typically be performedmuch more rapidly. Additionally, typical trapping frequencies for atoms are in the MHz or even sub-MHz range.If we can excite and de-excite atoms very quickly, the presence of atoms in Rydberg levels does not actually im-pact the trapping of the atom (even though Rydberg states are typically anti-trapped under the typical trappingfields!). By simply turning off the traps during Rydberg interaction, and turning the trap back on once complete,we cna continue to trap the atoms, since they do not have time to fly out of the trap.

Practical ApproachesIn principle, one can choose any atom (or even a molecule)! However, we generally want to choose atoms withrelatively simple structure, such as Hydrogen-like atoms (Alkali atoms) such as Rb, Na, Cs, which are quitestraightforard to control. At the next level of complexity, Alkaline-earth atoms such as Sr and Yb can be used.

Even after the atom is chosen, there are usually multiple choices of qubit encoding. The most straightforwardapproach is to use |g〉 and |r〉 as computational basis states, and manipulations are quite straightforward ingeneral. Although note here, we cannot necessarily access the full 2n dimensional subspace due to the blockade.Another approach is to use two ground levels split by the hyperfine interaction, with a laser carefully selectedto only excite one of the qubit levels to |r〉. This has the advantage that the ground states can have excellentcoherence properties. A final option is to even use various different Rydberg atoms (e.g. of the s and p character).

As an example of control and entanglement, we can consider |g〉 and |r〉 as the qubit basis states. The idea isto use the Blockade to prepare a Bell state of two atoms in nearby tweezers (sitting within Rb). If the atoms startin |gg〉, and we drive the atoms to |r〉, we will be able to excite at most 1 atom. Now we will undergo oscillationsbetween |gg〉 and a state where only one atom is in the Rydberg state. However, since we cannot fundamentally

62

determine which of the atoms is excited, we effectively drive |gg〉 |W 〉 ∝ |gr〉+ |rg〉, where the phase is set by therelative phase of the laser between the two atoms.

This is partially evident in the oscillations in the probability to detect 0 and 1 atom in the Rydberg state.However, such oscillations are not sufficient to tell whether or not we have created an entangled state. In orderto show that we have entanglement, we need to be able to measure the relative phase between the components,not just the overall populations. This is equivalent to distinguishing |Ψ+〉 from |Ψ−〉. In order to do this inthe experiment, we can add a differential phase shift on one atom relative to the other by application of anadditional laser field (via the AC stark effect). If this phase is applied when the atoms are in the entangled state,we transform to a state like |ge〉 + eiφ |eg〉. Now, for example if we are in the state |D〉 ∝ |ge〉 − |eg〉, the laserwill no longer have the correct phase to de-excite back to |gg〉. And thus, oscillations in the signal with respectto |φ〉 directly probe this phase, and allow one to extract entanglement fidelity.

This procedure creates the entangled state desired, but does not show us how to implement universal gateson this two-qubit register. This can be seen by the fact that |rr〉 cannot be accessed by such excitation. Thesolution to this is to change the qubit basis. The basis |g〉 , |r〉 is not ideal for a few reasons. First, it has a finitelifetime ∼ 200µs for the Rubidium 70S level, for example. Additionally, Rydberg atoms are not trapped, leadingto some loss. And finally, as already mentioned, we cannot access the full Hilbert space.

Instead, if we use qubits based on hyperfine spin states, we can solve these problems. Now, we can usemicrowave fields to fully manipulate the ground state qubit, which has excellent coherence times. Then, toperform multi-qubit gates we need to perform atom specific light-shifts via the Rydberg levels. One naturalapproach is to perform a CZ-type gate of the form

|00〉 → |00〉|01〉 → |01〉 eiφ

|10〉 → |10〉 eiφ

|11〉 → |11〉 ei(2φ−π).

(259)

Here, by choosing now a non-zero detuning for the Rabi excitation pulse, we can pick the detuning and pulseduration such that the atom undergoes a full oscillation on the Bloch sphere (not necessarily by passing through|W 〉, since we now have some detuning. Now, the area enclosed by the trajectory will be the phase acquired bythe state. By choosing ∆ appropriately, we can implement (259). By combining this technique with individuallight-shifts to impart sigmaz rotations, one can perform quantum logic operations on small registers. Further-more, the same idea can be used on three qubits within Rb simultaneously to implement three-qubit gates directlywith just three atoms - no need for auxiliary qubits. This is helpful, since it allows one to directly implementcomplicated multi-qubit unitaries without overhead, related to the idea of CNZ gates discussed earlier. Finally,these operations can be performed on several sets of atoms in parallel. For details see H. Levine et al, PRL (2019).

Quantum simulationsThe beauty of the neutral atom approach is that we can control many atoms in parallel. This is a nice setting forapplications involving quantum simulation. Historically, this idea is the origin of the field of quantum informa-tion. Richard Feynman pointed out that simulating quantum systems is hard, and that we should use quantumhardware with programmable interactions in order to simulate the physics.

This can be done in the neutral atom case by assembling atoms in configurations equivalent to effective-spinsystems, and simulate the dynamics. We can understand this by looking at the many-body Hamiltonian givenby

H =∑i

hΩ

2σix −

∑i

h∆ni +∑i<j

Vi,jninj (260)

for ni = 0, 1 are Rydberg atoms on each cite i. This is very faimilar to the Ising Hamiltonian, except for thisterm adding an energy Vi,j . Note that we can effectively replace n = 1+Z

2 . The middle term, proportional to ∆,the laser detuning, can determine whether all atoms are in the ground state or in the excited Rydberg state, inthe absence of other terms. If we now add itneraction such that we have nearest-neighbor blockade, we can onlyaccess states with so-called Z2 ordering, with every other atom excited. By increasing interaction range, we seedifferently ordered state Z3, Z4, etc (every third, every fourth atom excited, etc). Here Zi refers to the brokensymmetry in the configuration of atoms in the ground and Rydberg states.

To explore these phases, you could start with all atoms in the ground state, and slowly change the detuning toenter the relevant phase adiabatically. By doing this for various interaction ranges, one can map the relevant phase

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diagram. The important concept is that the system undergoes a phase transition. This is special, since we aredealing with an isoltaed system with effective temperature of zero. This means that the phase transition is drivennot by thermal fluctuations (as with typical statistical mechanical phase transitions), but rather by quantumfluctuations, making these quantum phase transitions. In the intermediate detunings, the system struggles todecide which phase to be in. This type of experimental system can probe snapshots of the physics of thesequantum phase transitions.

This can now be done in larger arrays. Once we are in the final phase, we typically see long, ordered domains,with domain walls or breaks in the individual domains. This is akin to a crystal boundary or dislocation. Bylooking at the domain wall density, one sees that the mean of the domain wall density varies smoothly. However,the variance of this domain walls is sharply peaked at the phase transition, and this peak signifies the quantumphase transition, as it is a result of the frustration of the system between the two phases. Finally, one can lookat the probability for a given microstate to occur (e.g. the ideal phase with no domain walls). This is a sort ofmeasure of how good the system is - not as quantitative as a fidelity - but as a measure of how probable we areto reach the exact ground state. For details, see H. Bernien et al, Nature (2017).

This is now a frontier of quantum computing and simulation. Extending these results to arrays of two-dimensional atoms with improved coherence using hyperfine qubits with individual atom control are the crucialnext steps. By using a combination of a spatial light modulator and an acousto-optic deflector to make arbitrarytwo-dimensional patterns, 2D experiments can now load on the order of 300 sites. One can verify the analogousphase diagram for the 2D Ising model. The first phase, for example, will be the checkerboard phase, the firstsignatures of which can now be seen.

The two dimensional patterns of atoms may also enable implementation of QAOA in a regime that cannot becompared classically otherwise. This is closely related to the idea of co-design, using a specific hardware systemto efficiently encode a problem. In this case, the problem is to find the maximum independent set (MIS) ofvertices on a graph such that no elements of the set are neighboring on the graph. An example of this, suitedto the Rydberg array, is the MIS on a unit disk graph where vertices are connected if they are within a givendistance. This is still an important problem in designing networks, and is still an NP complete problem. Thegraph now corresponds to the ground state of the system, since we are trying to excite as many Rydberg atomsas possible, given the constraint of the Rydberg blockade. Thus, by finding the ground state, e.g. by QAOA,one can solve this problem. Of course, QAOA is a heuristic algorithm, so there is no guarantee that it will workwell. Now, the challenge is to benchmark algorithms, such as QAOA, against the best classical algorithms.

Outlook: There are now a handful of atom-array experiments with different flavors. Simultaneous, individ-ual control corresponds to having large numbers of independent laser beams with rapid modulation control is amajor challenge. Increasing both atom number and circuit depth is also a formidable challenge, since individualerrors on the circuit or atomic level become exponentially important as the system size is scaled up. To solvesome of these problems, techniques such as Rydberg atom trapping and control of spontaneous emission may benecessary. Additionally, making fast measurements is key to atomic and ionic approaches to quantum comput-ing. These emeasurements are currently the slowest part of the computation, so this limits the amount of timerequired to perform an algorithm. Finally, identifying the useful algorithms and approaches is an open question.Building error-correct systems that can perform arbitrary quantum algorithms is also an outstanding goal.

4.4 Superconducting quantum computer

Our description of superconducting qubits goes back to harmonic oscillators, this time of a circuit made of acapacitor and an inductor. LC circuits have a resonance frequency at ν = 1/

√LC, coming from the differential

equation describing the charge. The energy stored in the circuit is E = 12LI

2 + 12Q2

C . We consider the flux

through the conductor Φ = LI as a dynamical variable such that E = 12

Φ2

L + 12Q2

C . We can therefore express flux

and charge in terms of creation and annihilation operators, with zero point motion Qzpf = h2Z , where Z =

√LC

is the ‘impedance’ which has units of resistance. This almost comes out to be 50-100 ohms. Comparing thisHamiltonian with harmonic oscillators, we can define operators Φ, Q as conjugate variables analogous to x andp such that [Φ, Q] = −ih and H = hν(adaggera + 1/2), where QiQzpf (a − a†) and Φ = φzpf (a + a†). Typicalparameters involve ν = 10 GHz and kT << hν, Vzpf ∼ µV and Izpf ∼ nA. Two problems emerge: 1. theresistance introduces loss, and the LC circuit is harmonic (linear).

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The approach to solve these problems is:

• Use superconducting elements that have R→ 0 when the temperature T < Tc.

• To introduce a non-linearity we use the Josephson effect, which acts as a nonlinear inductor.

What are each of these things physically? An example is to use a superconducting (sc) island separatedfrom ground (also sc) via a thin insulating film. The island has charge Q. Provided that the insulator is thinenough, what happens is that there is a non-negligible probability for charge from the superconductor to tunnelacross the potential barrier. Classically, the barrier has infinite resistance, but there will be some probability oftunneling. The physics is the following. The fundamental reason why sc have zero resistance is the energy gap.Inside sc, electrons which are usually fermions effectively attract each other and create cooper pairs, at least atlong range. They form these objects from two fermions (2e), but the new pairs are actually bosons, such thatthey can condense in the ground state. These cooper pairs occupy energy states which are gapped from the restof the spectrum, where the gap is related to the binding energy of the electrons in the cooper pairs. Only finiteenergy is required to break these cooper pairs (corresponding to Tc), but below that, they can condense intothis state and behave like a condensate. In this state they can undergo motion without friction at all, namelysuper-fluidity, and that is what gives rise to superconductivity. These electrons travel in pairs, such that whencharge tunnels through the barrier it happens in pairs. As long as the gap is much larger than the thermal energy,only the bottom state can be occupied on both sides of the insulator, such that the pairs can tunnel across thebarrier coherently. The tunneling Hamiltonian is given by:

HT = −1

2EJ∑m

|m〉 〈m+ 1|+ h.c. (261)

where n = N +m and n is the number of cooper pairs in the island. Note that the eigenstates are not Fockstates |n〉! The eigenstates are |φ〉 =

∑+∞k=−∞ eikφ |k〉, namely:

H |φ〉 = −Ej cosφ |φ〉 (262)

and the physics is that φ is the phase of the sc wave function on the island. The cooper pairs all occupya macroscopic quantum state which has a phase and amplitude, and we have written φ as the phase here. Inparticular:

n |φ〉 = −i ddφ|φ〉 (263)

such that in some way, n the number of particles acts as the momentum degree of freedom for the position,and the Hamiltonian is:

H =Q2

2C− Ej cosφ (264)

When we write this in terms of the phase of the wave function, we see that the tunneling energy is a nonlinearfunction of the phase. The cosine is ≈ Ejφ

2/2 for small phase φ, and we recover the harmonic oscillatorHamiltonian. Following the canonical commutation procedure, we have:

H =Ej2φ2 + 8Ec

n2

2[φ, n] = −i

H = hj(a†a+ 1)

ωj =

√Ej8Ech

φ = φzpf (a+ a†)

φzpf =

√2EcEj

(265)

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the an-harmonicity is Hah = −Ejφ4/4! = −Ec2h (a†a†aa + 2a†a). We call EC the charging energy. The spec-trum of the system is two lowest state separated by the plasma frequency ωj which is a few GHz, and thenthe third excited state is at a separation of ωj + EC , where the anharominc term Hah ∼ EC . Fundamentallywhat gives rise to this term is the coulomb interaction. You can think about this problem as a linear oscillator+ nonlinear Josephson Junction, OR we can say let’s neglect all interactions and look at linear motion of theparticle. Tunneling is in principle a non-interacting particle (linear) effect. If you neglect the Q2 term, all wewill have is linear motion of the quasi-particles. Then adding the charging term, there are corrections from thecharging energy ∼ Q2 which introduce a non-linearity.

Remarks

1. The derivation above is kind of sloppy. In one way, φ is NOT an operator, but eiφ is. The phase should beperiodic. The way we have derived things above loses the periodicity. We can do it the way we did abovesince we assumed that the phase fluctuations are small.

2. The effect of an external E field by applying a voltage across the gap is that we. must add another term tothe Hamiltonian: H → H − V0Q = 4e2(n− n0)2/2C ≡ EC(n− n0)2. The n0 = CV0/2e is the equilibriumcharge sensitive to V0. This term is both good news and bad news. On one hand, we can use voltage tocontrol the system, which is nice (we control classical computers with voltages!). However it turns out thefluctuations introduced by this term is parasitic. In practice, we would like to minimize the sensitivity ofthe qubit to charge noise by minimizing n0 without losing nonlinearity.

3. We would like to work in the optimal regime where EC << EJ but finite EC for nonlinearity. This is theidea of the transmon qubit charge island such that EC/EJ ∼ 1/50 or 1/100.

4. The typical best coherence time is T1 ∼ ms and T2 ∼ 10µs, limited by acharge in purities.

5. By adding a magnetic field, youc an efefectively control the background flux. In analogy with the simpleharmonic oscillator, we can re-write the phase φ as the flux φ = 2e

h Φ = 2π ΦΦ0

and Φ0 is the flux quantum.The physics of the eJJ is nonlinear inductance, such that adding a magnetic field controls the effectiveinductance L. Many variation of sc qubits exist and it is an ongoing field, but in all of them we usemicrowave fields to manipulate the qubits.

6. the superconductors are made out of niobium or aluminimum and the tunnel barrier is aluminum oxide(oxidized aluminum). The Schoelkopf group at Yale attached a large capacitance to one of the cooper pairboxes which acts as an antenna. The antenna is then used to control the qubits.

7. to operate the circuit quantum mechanically, the circuit must be cooled well below the critical temperature(typically to 0.01 K). Then the circuit is isolated from the environment by only connecting the circuitelectrically to the microwave line. Then we are left with a nonlinear oscillator where the qubit can beencoded in n = 0 and n = 1 states.

The remaining issue, is how do you coule qubits and perform multi-qubit gates? The idea is to employ coupledoscillators. One qubit is coupled capacitively to a linear LC circuit which is used as a data bus. Then qubit 2 iscoupled to the same linear LC circuit. The physics is that of two coupled oscillators such that HQ−res = Vg2enwher Vg is the voltage on the coupling capacitor created by the LC circuit. The voltage acts as a driving voltagefor the qubit, such that the two oscillators are coupled through normal electric potential Vq. Therefore, thequantum degrees of freedom of the linear circuit can be coupled to the qubit, when we write Vg = Vzpf (b+ b†).One popular linear resonator that people use is a stripline about 25 mm long with a 2µm gap. The excitations ofthe stripline are microwave photons inside the circuit, which are confined to a volume ∼ λd2 where d is the gapin conductors of the waveguide and λ is the wavelength of the microwave field and the length of the waveguide.

the caivty and circuit QED systeem is dseecribed by a TLS and SHO. For g << ν:

H = hg(b+ b†)(σ + σ†) ≈ hg(bσ+ + b†σ−) (266)

such that we arrive at the same JC Hamiltonian as we saw before with trappeed ions. The physics of g is:

• Suppose that the qubit is a dipole moment d (the big antenna with the capacitor attached)

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• The coupling g is the dipole moment dE0/h where E0 is the electric field of one microwave photon in hteresonator.

• The electric field from one photon is given by hν = ε0∫|E0|2dV = ε0|E0|2V where the volume V is the

volume of the e/m field.

• therefore g = d√

hνε0V

/h. The quantity g then decribes how much one single ephton in the resonator affects

the qubit. The typical values are g > 100MHz.

To use the total circuit (circuit QED) we tune the resonator into resonace with teh qubit which allows forcoherent exchange of excitations, and generate nonclassical states of MW radiation. However the most commonway to use it is to introduce a detuning between teh resoantr and the qubit that is much larger than g. In thiscase, using second order perturbation theory we see that:

Heff =hν

2Z + hωrb

†b+ hg2

ν − ωrb†bZ (267)

We can think about this Hamiltonian in two ways:

1. The qubit has a refractive index which the resonator experiences, and can be used for the qubit readout.

2. There is a photon dependent shift in the qubit frequency. The application for that is a two-qubit gate. Forexample, for two qubits coupled to the resonator, we can map the state of qubit 1 to the resonator modeand then it will be sensed by qubit 2.

For qubit readout, there is a shift in the frequency of the microwave resonator, such that by detectingtransmitted or reflected photons the qubit state can be read out. There are many groups in academia andindustry pursuing these approaches. There are several challenges and opportunities:

• The lifetime and coherence times of the qubits is limited to 10-100µs for almost a decade because of chargedefects in the JJ, as well as imperfections in the sc. This is because even at low temperature, sc still havesome quasiparticles (people don’t understand where it comes from and it’s an area of current research).

• Using transmons that are 3D and much larger cavities have much longer coherence times than qubits, butthey are hard to scale. Each qubit is a centimeter scale!

• All qubits are different - there is some inhomogeneity between qubits from fabrication imperfections, whichdramatically increases the complexity of the control needed.

These motivate new approaches for scaling (for example, using bosonic degrees of freedom to store informa-tion), as well as new approaches to quantum error correction. For further references see S. Girvin’s lecture noteesand arXiv 1302.5842; M. Devoret & Shoelkopf Science 339, 1169.

In summary, for implementation of QC’s there are contradictory requirements: isolation from the environment,but control of qubits and their interactions. The state of the art of gate fidlities is F = 0.99 − 0.995. For an Nqubit machine, running a p-circuit depth algorithm, then you find that FNP << 1 for NP ∼ 100− 1000. Thereare two approaches to solve this problem: increase fidelities, or apply quantum error correction.

5 Quantum error correction

We have shown that most of the quantum channels we’ve discussed can be represented as random Pauli matricesacting on the qubits. These are the kind of errors we need to correct. In practice, there are two types of errors:(1) memory errors, where the quantum state is not maintained due to coupling to the environment which can beaddressed with QEC, and (2) operation errors, which will be reduced to memory errors.

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5.0.1 Classical error correction: review

One simple way to correct classical error is to use redundant coding. Suppose that we have a single classical bitwe want to store for time t. However, a bit flip error occurs with probability p. The idea of EC is to encodeone logical bit into several physical bits. For example, we can encode 1L as 111 and 0L as 000. Then, considerthe bits 0L after time t. The outcomes will be 0L with probability (1 − p)3, but there will be one bit flippedwith probability 3p(1 − p)2, two bits are flipped with probability 3p2(1 − p) and three bits will be flipped withprobability p3. How can we correct this error? If the p << 1, then the errors will be dominated by a single bitflip nature. We can then measure all qubits after time t and if the majority of them are in 0, we assign 0L andflip the bit in 1. This is called majority voting.

Remark

1. This procedure will work only if a single bit is flipped. For two or three errors, the procedure will fail.

2. The probability of a correct outcome is then (1 − p)3 + 3p(1 − p)2, such that we have removed the linearterm in p, such that this procedure is helpful if p < 1/2. This makes sense: if p > 1/2, most of the timetwo or three bits flip.

3. To store for a long amount of time, divide the total time into shorter amounts of time τ , and do EC aftereach τ = t/N . For large time N , the probability of the bits being in the correct state after error correctionis p ≈ (1− 3(ct/N)2)N , such that N >> 3(ct)2

5.1 QEC key idea

We can use redundant encoding in quantum states such that |0L〉 = |000〉 and |1L〉 = |111〉. There are twoproblems: (1) how do we measure without destroying superposition? And (2) the errors are continuous, suchthat one superposition state is changed to another.

To start we consider single qubit errors: the bit flip σix = Xi, similar to the classical case, as well as a phaseerror σiz = Zi. Let’s treat them individually.

5.1.1 Bit flip error

Consider three physical qubits such that |0L〉 = |000〉 and |1L〉 = |111〉 and a general state |ψ〉 = c0 |0L〉+ c1 |1L〉.The outcomes are |ψ〉 → |ψ〉 with probability (1− p)3 a single bit is flipped with error 3p(1− p)2, etc.

For a correction, we consider a collective measurement of four operators:

P0 = |0L〉 〈0L|+ |1L〉 〈1L|Pi = σixP0σ

ix

(268)

where i runs over the Pauli matrices. This has the following properties: P0 |ψ〉 = |ψ〉, Pi |ψ〉 = 0, Piσjx |ψ〉 =

δijσix |ψ〉, such that Pi projects into σix |ψ〉. The logical state is an eigenstate of the first projector P0, and the

other projectors identify which bit has been flipped! With the measurement result, one can then proceed to fixthe error. Also, this procedure works regardless of c0 and c1: it doesn’t matter what the superposition is. Thekey is to do a measurement on the redundantly encoded state, without revealing what the state is.

The procedure is to measure P0 and if it is 1, the state is fine, if it is 0, then measure P1. If that measurementis 1, apply σ1

x and if 0, move on to P2.Remarks

• This works only if a single bit has flipped. If two or three bits was flipped, then there is a problem.

• After the corretion we have |ψ〉 with probability (1− 3p2 + 2p3) such that the linear term is gone.

• MISSED THIS REMARK

• What if the error was coherent? For example what if |ψ〉 → |ψ〉+ ησix |ψ〉? In this case, measuring Pi stillworks! For example, if Pi = 1, then it forces the system into σix |ψ〉! A measurement of P0 = 1 projects thestate into |ψ〉. The measurement digitizes the coherent, continuous errors! This is very important.

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• The physics is that the measurement projects into different subspaces of the Hilbert space. It either projectsyou into the logical subspace HL or out of it, into an orthogonal subspace, which can then be corrected -you know how these states are related to the original states in HL.

• How do we measure these P operators in practice? We want to distinguish e.g. |000〉 from |100〉. One wayto do this is to measure Z1Z2 and Z2Z3. If we measure both operators on the logical states, we alwaysget an eigenvalue of 1: these operators compare the first and second qubit, and the second and the thirdqubits. However, if the measurement is on the state |100〉 or a state where one bit was flipped, this will nolonger be the case. In particular a measurement of ZiZj compares the i and jth qubit.

5.1.2 Phase errors

In quantum mechanics, there are also phase errors! How can this procedure identify those? They correspond toacting the Z Pauli matrix. We can recognize that a bit flip error in one basis is a phase flip error in another, andvice versa. We can use XiXj operators therefore to correct phase flip errors. For a state |ψ〉 = c0 |0L〉+c1 |1L〉, thephase flip error brings it to |ψ〉 = c0 |0L〉− c1 |1L〉. We can now measure in the X basis, noting that σz |+〉 = |−〉and σz |−〉 = |+〉. To protect from Z errors we can encode |0L〉 = |+ + +〉 and |1L〉 = |− − −〉.

However, what if we have both errors - for example, a depolarization channel?

5.1.3 Shor’s 9 qubit code

The trick is to combine protection from σx and σz errors by using the states:

|0L〉 =1

23/2(|000〉+ |111〉)⊗ (|000〉+ |111〉)⊗ (|000〉+ |111〉) |1L〉 =

1

23/2(|000〉 − |111〉)⊗ (|000〉 − |111〉)⊗ (|000〉 − |111〉)

(269)

How does this work?

• if there’s a spin flip, e.g. σ1x, then a measurement of σ1

zσ2z will be −1, and a measurement of σ2

zσ3z is 1.

• For a phase flip σiz |ψ〉 we can measure X1X2X3X4X5X6, X4X5X6X7X8X9. These measure if the phase isthe same on the first three set of qubits as on the second three sets of qubits, and likewise on the secondthree set of qubits on the last three set of qubits. In this case, if the phase of the first qubit flips, then thephase of the first block will not be the same on the second block, etc. Then you can apply a phase gate toany of the qubits in the block that has the error. The interesting thing here is that the same effect happensif the phase of the second or the third qubit in the first block has flipped and is corrected.

Why does this work? The GHZ states are eigenstates of the X operators! Namely, σixσjxσ

kx(|000〉 ± |111〉) =

±(|000〉 ± |111〉), such that a measurement of X1X2X3 is kind of like a measurement Z1 and X4X5X6 is like ameasurement Z2 in a rotated basis where the ± states are the Z basis states.

The bottom line is that if we perform measurements of these six operators as well as pairs of the ZiZj ’s (6products of two operators), we can correct ALL single qubit errors! However, still we cannot correct two qubiterrors or more. The resulting total error after QEC is the following: for a bit flip error it is 3p2x3 blocks which is9p2, and for a phase flip error it’s 3p2 probability of error, such that the total error is 12p2. This procedure onlyimproves if 12p2 < p. The reason why this error grows is that we had to use more qubits in a more entangledand complicated state.

5.1.4 Implementing QEC

How to measure σ1x...σ

6x The basic idea is to use an additional qubit ‘ancilla’ |ψa〉 and prepare it in a superposition

|0a〉+ |1a〉. Then perform CNOT gates with the ith qubit: Uaic |ψa〉 |ψi〉 = 1√2(|0〉a |ψi〉+ |1a〉σix |ψi〉. Hence after

6 CNOTs we have the state |0a〉 + Π6i=1σ

ix |1a〉 on the ancilla qubit, which we can then measure in the x basis.

You can also use the same procedure to measure products of Z. In total, we need 24 CNOT gates - this is a lot!Additionally, the operations on the encoded states must be collective. For example, to flip from |0L〉 to |1L〉

- such that these operations must be collective.

Remarks

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• The generalization of Shor’s code is the stabilizer formalism. For example, a 3bit code is S1 = Z1Z2, andS2 = Z2Z3. This divides the Hilbert space into logical segments, namely no error, and each of the singlequbit bit flip errors. One can generalize this using abelian subgroups S with some elements [Si, Sj ] = 0, suchthat they simultaneously diagonalize the whole Hilbert space H2n . One can say that |ψ〉 is stabilized by Sif for all Si in S, Si |ψ〉 = |ψ〉. For S with l generators, each subspace labeled by S1, ...Sl has dimensions2n−l which requires n− l qubits. For further reading see Nielsen & Chuang.

• Operational errors introduce two kinds of problems: the error propagate and multiply, for example, a CNOTgate with an error on a control qubit brings the error to the target qubit. Moreover, if the CNOT gatesthemselves are imperfect, they can also introduce errors, in principle on both of the qubits simultaneously.

• The bad news is that if a circuit creates two errors σixσjx there will be no recovery (the code only corrects

single qubit errors).

• The good news is that operational errors can be treated as memory errors - we can assume the gates areperfect, but that after each gate we need to apply some memory error correction. This bit of good newscan be used to develop the idea of fault-tolerant quantum computing.

The idea of FTQC is to build circuits in a clever way such tath in each step in the circuit you introduce atmost one error into each logical encoding block. If this happens, those errors can be corrected at hte next step.You exchange QEC step with logical gates to correct these. If p is the probailiy of failure of each individualcomponent CNOT, then the evolution can be constructed carefully enough such taht the probaility oto intrudcetwo errors ias at most p2.

5.1.5 Example: FTQC

A FT CNOT gate is to start with two logical qubits (each having 9 qubits). First do QEC on teh input controlstates, then do CNOT, then QEC on the output states. What are the possiblities when 2 errors are introducedin one qubit?

1. TEH QECC code error simultaneous on 1 and 2: 24 CNOTs for 1 step of QEC if one fails with probabilityp1, the total probabiltiy of error in both upper and lower logical qubit is (24p1)2.

2. If there is an error in one block and a failed CNOT, that will also introduce 2 errors. The total errorprobability will be 2(24p1)(100p1) where 100 comes from the number of CNOTs from a block CNOT.

3. One error correction step fails has probaiblity (24p1)2 this will also intrduce two errors.The total probaiblity of having two errors after QEC and CNOT is 103− 104p2

1. We require that this error issmaller than the probaility fo a gate error in one qubit. If this is true, then the two error rates do not increaseoverall and p1 < pth = 103 − 104. The probability pth is the threshold error. This is the basis for FTQC.

REmarks:1. Very important result: can keep the error from magnifying if p1 < pth!2. The perfect decoding at teh end of computation – the resulting error will be p2

1

3. Cascaded encoding can be used to reduce error even further to do p2m for m levels of encoding or numberof cascades. See Nielsen & Chuang or Preskill’s notes for further reading.

4. The error threshold is a peculariar thing! Specifically, it depends on the details of QECC, and it assumesthat I only have errors in (GET THIS)? It also assumes that the measurements are efficiently made with time,and that it assumes the errors are completely uncorrelated on all qubits - the noise is uncorrelated.

Teh types of coupling e.g. nearest-neighbor or all-to-all matters. The exact pth is not known, and currentreasonable estimates guess that pth ∼ 10−3 − 10−6.

It is extremely unlikely that any practical QC can be built using these existing methods in Misha’s opinion.Directions to explore include:

1. More efficient codes lower pth + ‘natural topology’2. Realistic situations and certain kinds of errors dominate (bias-preserving oeprations) - certain operations

are difficult.3. Other approaches include e.g. qubits protected by physics and topology. See for example A. Kitaev’s

papers, and MS project.4. More efficient approaches for specific architecture: cat codes, colective gates etc - this is a current frontier

of current research! This is the idea of co-design.

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6 Quantum complexity theory

This will be a brief single guest lecture by Boaz Barack. Two great references to learn more: Math and Compu-tation by Avi Wigderson and Quantum Computing since Democritus by Scott Aaronson.

The first question in discussing computational complexity we must ask is what do we want to solve? The twotasks we will discuss is the computation of a function with input x and we want to produce an output f(x). Iff is boolean, we sometimes call this “deciding a language.” The second will be to sample from a distribution.Given an input p,, whcih is some description of a distribution, the algorithm may use some internal randomnessbut the output x ∈R p. is sampled from the distribution.

In the case of the compute function, we can consider an input string x with size n, we perform a number ofoperations S(n) that will grow with the size of the input, and we output f(x). Note that we can formally define(using Turing machines or boolean circuits) the function

f : 0, 1∗ → 0, 1 ∗ computableinS(n)steps. (270)

Examples

1. Multiply: a, b→ ab

2. Factor: m→ p1, ..., pk s.t. P1p2....pk = m

3. Min cut: G→ S to mimimize the number of edges

4. Max cut: G→ S to maximize the number of edges.

We can discuss the best known algorithm for each of these algorithms. Note that here we want the algorithmto work on any inputs, which includes the “worst case” inputs which may be the computationally hardest. Forexample, multiplication takes roughly n2 using standard algorithm (that we learn in grade school). Karatsuba’salgorithm can get n1.6, and the best is O(nlogn) in a way similar to the fast Fourier transform (since the FFTis an efficient way of effectively multiplying polynomials).

For factoring, the naive algorithm is to check all of the numbers (say, up to square root of the number tofactor). So it takes O(

√m). However, if the number of digits of m is n, then this will take roughly 2n/2 steps.

There are more clever algorithms such as the quadartic sieve and the nonlinera sieve, which lead to 2n1/3

, whichis better but still exponential in n. Min cut can be done in O(n2), and max cut is actually ≈ 2n, with no betterknown algorithm. This is interesting: it often happens that the complexity of minimizing an objective is mucheasier than maximizing the same function. These can be very different tasks.

In complexity theory, we can distinguish between problems of varying difficulty. The largest class are problemsthat can be solved in exponential time EXP (e.g. 2n, 2

√n. We call the problems solvable in polynomial time P.

Formally,P : fstS(f) ≤ a.nbforconsta, b (271)

This is the smallest class, and is a subset of of EXP. We call these problems ones that “we can solve efficiently.”Of course, depending on the details of the scaling it may or may not actually be efficient, but comparatively, wecna solve these problems since they do not scale out of control with the problem size. Next, we call problesmP ⊆ NP ⊆ EXP , in terms of set-theory. An example of NP is max cut. These are problems that we knowwe can verify in polynomial time. NP is in some sense, the problems that we actually want to solve, since wecould usefully verify the answer once we have a solution. Note that NP does not mean “not polynomial” - it onlymeans that the solution can be verified in polynomial time.

It is possible that P = NP ( EXP , but it is also possible that P ( NP = EXP . We don’t really know. Ofcourse, it is also possible (and strongly believed) that P ( NP ( EXP . In fact, this is usually widely believedin computer science and used as a working hypothesis, but we do not have a working proof of this.

So where does quantum fit in? It fits in the class BQP:

BQP : f |x→quantumcircuit f(x) in ≤ a.nb gates (272)

Since we know that classical gates can be performed with polynomial overhead on a quantum computer, we knowthat P ⊆ BQP . Since we know that classically we can simluate a quantum computer we know that ⊆ EXP .

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Again, it is not known strictly whether these are subsets or equalities, however it is believed P ( BQP ( EXP .For example, we know that factoring is in BQP, but it is not known whether factoring is in P. Note that thequantum fourier transform is the workhorse of problems that are known to be in BQP but are believed not tobe in P. There may be other ways to get exponential speedups, but it is not necessarily known. Also, it may infact be that BQP extends into EXP - this is also not known.

There is a notion of problems such as 3 SAT and MAX CUT which are known as NP-complete (NPC). It isconjectured that none of them are in P nor BQP (not certain, but widely believed). One reason for this is thatit can be shown that if even a single problem is in P, all problems in NP would be in P (or in BQP as well). Theformal statement of the theorem is that if MAXCUT ∈ P → f ∈ P ∀ f ∈ NP . The way of proving thisis so-called reduction, where we can take an algorithm for MAXCUT and translate it into an algorithm for f .In other words, if we have a magic box that solves MAXCUT, by using this magic box we could solve f . Notethat this theory (and overall description) does not necessarily apply to best or average cases of problems, but toworst case implementation.

The canonical way of thinking of a problem in NP is by considering a circuit which has input p1...pk, andthe output is 1 if p1...pk = m, and 0 otherwise. In fact, we can rephrase the factoring problem in terms of thiscircuit. This circuit can be taken and mapped onto a graph such that if there was an input that made the output1, there would be a set that cuts ≥ 0.9n of the edges. So we can reduce the task of this generic circuit intofinding the MAXCUT.

The correllary of this is that if we have a polynomial time algorithm for MAXCUT, we get a polynomial timealgorithm to solve any efficiently verifiable task. For example, suppose someone gave you an encryption tahttakes plaintext and a key, and outputs cipher text. We now the algorithm and the ciphertext and the plaintext.However, we don’t know the secret key. Of course, if someone told you the secret key you could verify (e.g.write a circuit that outputs 1 for the correct answer). In other words, if you could solve MAXCUT, you couldbreak all crypto-systems. This is to be distinguished to known quantum algorithms like Shor’s algorithm whichbreaks factoring based crypto-systems, but not necessarily all crypto-systems. In fact, we believe there are cryptosystems that take an exponential amount of time even on a quantum system, implying that BQP ( NP . Theconjecture is that any quantum algorithm requires 2n/2 operations, as opposed to 2n for classical systems. Anexmaple of this is Grover’s algorithm, which is actually known to be the best. One can make a corrolary to thisthat an NP problem will take of the order ≥ 2εn operations.

So how do we cope with NP hard problems? Recall that some problems may be NP hard in the worst case. Ifthe input is some (x1, y1)...(xn, yn), and the goal is to find f ∈ NN(d, S) ∀i f(x) = yi. Training a neural net NNis NP hard in the worst case (but not always). However, we know that we can do this efficiently for some usefulproblems. In the context of graph theory, we sometimes pick special graphs such as bounded graphs, graphs onthe edge of a genus, that can be solved with specific algorithms in polynomial time. These are algorithms forspecial instances. It is also possible for random instances (not necessarily well known in advance, but likely tooccur). Finally, sometimes we change the problem and get more data by using less parameters and modifyingthe problem until we can actually solve it. For example, we may want to solve equations on integers but couldsolve them on real numbers and use rounding.

Now, onto quantum supremacy. It is a slightly different computational task, since we are given a distributionand we want to sample from that distribution. It turns out that this is very closely related to the problem ofcounting, where the input is a function f : 0, 1n → 0, 1, and the output is to count

∑x∈0,1n f(x). For

example, instead of MAXCUT you could ask how many cuts of a certain size do we have. One example of adistribution we may want to sample from is p over an n × n random matrix A such that p(A) ∝ perm(A)2,referring to the permanent, which is defined as

perm(A) :=∑τ∈Sn

n∏i=1

Ai,τ(i). (273)

There exists a conjecture that no polynomial time randomized algorithm exists for computing the permanent.The ideology behind tackling these problems on a quantum computer is the following. Building a full-fledged

fault tolerant quantum computer is hard. Perhaps it is not as hard to build a quantum computer that can solvesome computational task which is hard for classical computers. Informally, the hope is that we have a quantumcomputer the size of roughly a refrigerator, that can perform a certain task that would take a classical datacenter the size of a small city. The hope would be that then as we scale up the size of the problem, the timeand resources for solving the problem on a quantum machine would scale manageably, and not exponentiallylike with the case of the classical machine. Here, we give up on the idea that the task is useful, just focusing

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on the fact that the task is hard, as an experimental demonstration that this technology can potentially provideexponential speedups.

The example for this is p described by a quantum circuit C. here we have

pC(x) ∝ ( 〈x|C|0⊗n〉 )2. (274)

The dream quantum supremacy experiment takes in a circuit, programs it onto the device, and we get out somex ∼ pC (a sample from the distribution generated by the circuit). In reality, the best we can hope is that givenC, we get x ∼ δpc+(1−δ)J where J ≈ (1−ε)numgates. (Here, J stands for junk). In practice, δ will be small, buthopefully not so small that after running the experiment a manageable number of times wen have a statisticallymeaningful number of samples that are not junk. In the case of the Google experiment, this was on the order ofa fraction of a percent.

There is strong evidence that the ideal sampling problem is hard classically, but not necessarily in the caseof the imperfect (δ < 1) circuits. In practice, another problem is that in the real device

p(x) = (1 + δ)2−n, (275)

comparing with p(x) = 2−n for x ∼ 0, 1n and p(x) = 3× 2−n for x ∼ pc. We would ideally have to verify theruntime as 2n classically to show that this is hard.

If this supremacy proof is true, the implications are as follows. We can actually provide quantum speedupsover classical algorithms. We have known this thanks to Shor’s theorem, but it is nice to see this experimentally.We can also think of this as an extension of Bell’s inequalities, which tell us that the universe cannot be simulatedby some classical machine with hidden variables.

Finally, consider the PCP theorem. Assume we have an efficient verifier which runs in polynomial time.However, we have some provers that are unbounded. The provers can compute f(x) and tell the verifier thatf(x) = 1. However, there should be some way of checking that the provers cannot cheat. If there is a barrierbetween the provers and we randomly interrogate the provers, even if f is computable only in exponential time,we can still verify in polynomial time. The point of the barrier is to force the prover to be non-adaptive. Inother words, we can now ask the provers subsequent questions which do not depend on the previous questionsand answers. So we can think of the provers responses as completely pre-defined (all 2n responses), and we queryat random and are guaranteed that we will get the right answer.

Suppose there exists a function H : 0, 1n → 0, 1. The question is does there exist an x such thatH(x) = 0? The PCP theorem tells us that it is possible to transform H(x) into a more robust form. If thereexists x such taht H(x) = 0→, there exists x such that the expectation over i, j, kHi,j,k(x) = 0. In other words,for all x such that H(x) > 0→ ∀ x the expectation value > 1/100.

This is related to recent work MIP∗ = RE

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